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so i know we talked about different pacemakers in the body , but i thought it 'd be fun to revisit that and show you an interesting example . so let 's start out by laying out the table we 'd set up before . we talked about the heart rate in beats per minute , and we talked about the heartbeat itself -- the length of the heartbeat , and we 'd measured the heartbeat in terms of seconds . and you remember , there 's a nice little relationship between the two of these , because if the heartbeat actually gets shorter , then you can have more heart beats in a minute . and so of course , then the heart rate goes up . so that 's a relationship that explains how it is that our heart rate goes up and down . and we talked about the sa node , the av node , and the bundle of his . and we said starting with the sa node , the heart rate was somewhere between 60 and 90 . and i think i 'd chosen 90 , just because that was a nice , easy number to do math with . and we had said that the heart beat is about 0.66 seconds . so that 's the length of a heartbeat there . and then we have the av node . i 'm just going to quickly go through this . i know this is recap for you if you 've seen the other video . if you have n't , then these numbers come from basically dividing beats per minute down into seconds . and so then each beat then would be one second for the av node . and finally , we did the bundle of his . and i think i 've started trying to take a shortcut in writing bundle of his into just boh . and that looks something like this . and those underlying numbers are the numbers i 'm using to calculate the heartbeat lengths . so that 's basically what we had come up with . and we had also talked about the idea of having delays . you actually need time for the pulse to be in transit basically . and so , i 'm actually going to add a third column to our little table here . and there really is no delay here , because the sa node is where things are starting . so let me actually just keep my colors the same . and then the av node , we know that there 's a small delay , because things do move pretty quick . so we said that here , it 'd be something like 0.04 seconds . so you can see that it 's actually pretty quick getting from the sa node to the av node . but then it gets even faster as you get down to the bundle of his . it actually takes only about 0.005 seconds . so it gets really , really fast . and remember that this transit speed , this is really related to conduction velocity . so how fast is the signal getting conducted ? so we call this conduction velocity . and the relationship between conduction velocity and the action potential is the slope of phase 0 . remember , the more steep phase 0 is , the faster something is going to go from cell to cell to cell . and actually , that brings up a good point , because in the av node , there 's a huge delay built in , because the conduction is so darn slow . and so you have to actually remember that there 's this 0.1 second delay . and generally speaking , i think of 0.1 seconds as almost nothing , but when you compare it to 0.005 seconds , because that 's the transit time -- that 's how long it takes the signal to get down , we said from the av node down to our particular bundle of his cell -- then all of a sudden , this delay is looking enormous . by comparison , this looks like a really big , big number . and let me just write transit here as well . so this is time for movement . and then the delay is simply getting through the av node itself . so this is all just rehashing what we 've talked about before . and finally , just to get at least a drawing down , because i like to draw , we have our sa node here . and we have our av node here . and we have our bundle of his over here . and let me draw it half the distance , somewhere like this . and remember , this is the direction of flow . we 're basically trying to move this way . and again , this way . so let me actually jump into something slightly new . so let 's assume for a second -- this is a thought experiment -- that instead of 0.04 seconds , i 'm just going to focus on these two right now . instead of 0.04 seconds , let 's say that it took 100 times as long . for some reason , let 's say that transit time for some reason , we do n't know why , let 's say it takes 100 times longer . so this ends up being 4 seconds , right ? 0.04 times 100 is 4 seconds . so let 's say it takes about 4 seconds , for some reason , to get a signal from the sa node to the av node . well , what would that mean for us ? what would that look like exactly ? and i think you 'll start seeing some interesting lessons from this little thought experiment . so , if that was the case , if it was actually taking about 4 seconds to get from one point to another , let 's now draw out a timeline . this is a little time line , and this timeline starts at 0 seconds . and then you have , let 's say , 1 second here , 2 seconds , 3 -- i 'm just going to see how far this goes -- 4 , 5 , and let 's go to 6 . so , this is 6 . seconds and we 're going to follow what happens over 6 seconds . so let 's imagine now we keep track of our sa node up here . and we 're going to keep track of our av node down here . so at time 0 , let 's imagine that everything is beginning . and we watch our sa node , let 's start with that one first . well , at 2/3 of a second -- because that 's about how long it takes , we calculated -- we would get our first action potential , or a heart beat would go through , right ? first beat . and that would then try to make its way towards the av node . so this one is going to try to make its way towards the av node . but we know it takes 4 seconds to get there . now , what happens after that ? well , you 'd have another beat let off . the first one has n't actually made it to the av node , but the second one is already done by that point . and you 'd have a third beat that goes through by that point . and so really , we 're counting these action potentials that are going through the sa node . and they just keep going through . they 're just going to keep flowing through here . and they 're going to all just continue and basically , just what are we going to get ? a total of probably 9 , right ? we 're going to get 9 signals sent off . now , take each of them is going to take 4 seconds to get to the av node . so when will this first one get to the av node ? this very first one will get to the av node somewhere around here , because that 's 4 and 2/3 of a second . so at 4 and 2/3 of a second , this one -- let me somehow show you without making this too messy -- will make it to the av node right here . of course at that time , the sa node itself is letting out its seventh action potential , but that very first one will get there at that point . now the av node , is it going to sit around and wait for 4 and 2/3 of a second to just go by and not do anything ? no way , right ? there 's no way , because what it 's going to do is it 's going to say well , let 's wait for a signal from the sa node . and at this point , it 's going to say well , nothing arrived from the sa node , so i 'm going to let off my own signal . and it 's going to keep doing this . so it 's going to go on its own rhythm now . so 2 , 3 , so all this time , the av node is on its own rhythm . and then finally , before av node is able to fire off its own fifth action potential by itself , a signal arrives from the sa node , this red arrow that i drew in . and so it 'll say , oh wait . we just got some positive ion passed through the electrical conduction system . so let 's go with it . so it 'll have a signal there . and then now , it 'll have another one here , because what happens at that point ? well , you have this guy arrives . he took 4 seconds , and he arrives right there . and then this guy is going to arrive after that . he 's going to arrives right there . so you see they start arriving . and so , once they start arriving , then you get back onto what looks like a normal rhythm . and so , it 's interesting because you basically , as a result of this long delay , have a phenomenon where for awhile , the av node is doing its own thing over here . and then after that , the sa node catches up . and then it continues on what would look like a normal sinus rhythm . and so , sometimes you 'll hear the term escape beats or escape rhythm . and so that 's what these are , these are escape beats , meaning they have escaped the normal flow of electrical conduction , which starts with the sa node . so , hope that was helpful .
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this very first one will get to the av node somewhere around here , because that 's 4 and 2/3 of a second . so at 4 and 2/3 of a second , this one -- let me somehow show you without making this too messy -- will make it to the av node right here . of course at that time , the sa node itself is letting out its seventh action potential , but that very first one will get there at that point .
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are the thresholds somehow raised ?
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so i know we talked about different pacemakers in the body , but i thought it 'd be fun to revisit that and show you an interesting example . so let 's start out by laying out the table we 'd set up before . we talked about the heart rate in beats per minute , and we talked about the heartbeat itself -- the length of the heartbeat , and we 'd measured the heartbeat in terms of seconds . and you remember , there 's a nice little relationship between the two of these , because if the heartbeat actually gets shorter , then you can have more heart beats in a minute . and so of course , then the heart rate goes up . so that 's a relationship that explains how it is that our heart rate goes up and down . and we talked about the sa node , the av node , and the bundle of his . and we said starting with the sa node , the heart rate was somewhere between 60 and 90 . and i think i 'd chosen 90 , just because that was a nice , easy number to do math with . and we had said that the heart beat is about 0.66 seconds . so that 's the length of a heartbeat there . and then we have the av node . i 'm just going to quickly go through this . i know this is recap for you if you 've seen the other video . if you have n't , then these numbers come from basically dividing beats per minute down into seconds . and so then each beat then would be one second for the av node . and finally , we did the bundle of his . and i think i 've started trying to take a shortcut in writing bundle of his into just boh . and that looks something like this . and those underlying numbers are the numbers i 'm using to calculate the heartbeat lengths . so that 's basically what we had come up with . and we had also talked about the idea of having delays . you actually need time for the pulse to be in transit basically . and so , i 'm actually going to add a third column to our little table here . and there really is no delay here , because the sa node is where things are starting . so let me actually just keep my colors the same . and then the av node , we know that there 's a small delay , because things do move pretty quick . so we said that here , it 'd be something like 0.04 seconds . so you can see that it 's actually pretty quick getting from the sa node to the av node . but then it gets even faster as you get down to the bundle of his . it actually takes only about 0.005 seconds . so it gets really , really fast . and remember that this transit speed , this is really related to conduction velocity . so how fast is the signal getting conducted ? so we call this conduction velocity . and the relationship between conduction velocity and the action potential is the slope of phase 0 . remember , the more steep phase 0 is , the faster something is going to go from cell to cell to cell . and actually , that brings up a good point , because in the av node , there 's a huge delay built in , because the conduction is so darn slow . and so you have to actually remember that there 's this 0.1 second delay . and generally speaking , i think of 0.1 seconds as almost nothing , but when you compare it to 0.005 seconds , because that 's the transit time -- that 's how long it takes the signal to get down , we said from the av node down to our particular bundle of his cell -- then all of a sudden , this delay is looking enormous . by comparison , this looks like a really big , big number . and let me just write transit here as well . so this is time for movement . and then the delay is simply getting through the av node itself . so this is all just rehashing what we 've talked about before . and finally , just to get at least a drawing down , because i like to draw , we have our sa node here . and we have our av node here . and we have our bundle of his over here . and let me draw it half the distance , somewhere like this . and remember , this is the direction of flow . we 're basically trying to move this way . and again , this way . so let me actually jump into something slightly new . so let 's assume for a second -- this is a thought experiment -- that instead of 0.04 seconds , i 'm just going to focus on these two right now . instead of 0.04 seconds , let 's say that it took 100 times as long . for some reason , let 's say that transit time for some reason , we do n't know why , let 's say it takes 100 times longer . so this ends up being 4 seconds , right ? 0.04 times 100 is 4 seconds . so let 's say it takes about 4 seconds , for some reason , to get a signal from the sa node to the av node . well , what would that mean for us ? what would that look like exactly ? and i think you 'll start seeing some interesting lessons from this little thought experiment . so , if that was the case , if it was actually taking about 4 seconds to get from one point to another , let 's now draw out a timeline . this is a little time line , and this timeline starts at 0 seconds . and then you have , let 's say , 1 second here , 2 seconds , 3 -- i 'm just going to see how far this goes -- 4 , 5 , and let 's go to 6 . so , this is 6 . seconds and we 're going to follow what happens over 6 seconds . so let 's imagine now we keep track of our sa node up here . and we 're going to keep track of our av node down here . so at time 0 , let 's imagine that everything is beginning . and we watch our sa node , let 's start with that one first . well , at 2/3 of a second -- because that 's about how long it takes , we calculated -- we would get our first action potential , or a heart beat would go through , right ? first beat . and that would then try to make its way towards the av node . so this one is going to try to make its way towards the av node . but we know it takes 4 seconds to get there . now , what happens after that ? well , you 'd have another beat let off . the first one has n't actually made it to the av node , but the second one is already done by that point . and you 'd have a third beat that goes through by that point . and so really , we 're counting these action potentials that are going through the sa node . and they just keep going through . they 're just going to keep flowing through here . and they 're going to all just continue and basically , just what are we going to get ? a total of probably 9 , right ? we 're going to get 9 signals sent off . now , take each of them is going to take 4 seconds to get to the av node . so when will this first one get to the av node ? this very first one will get to the av node somewhere around here , because that 's 4 and 2/3 of a second . so at 4 and 2/3 of a second , this one -- let me somehow show you without making this too messy -- will make it to the av node right here . of course at that time , the sa node itself is letting out its seventh action potential , but that very first one will get there at that point . now the av node , is it going to sit around and wait for 4 and 2/3 of a second to just go by and not do anything ? no way , right ? there 's no way , because what it 's going to do is it 's going to say well , let 's wait for a signal from the sa node . and at this point , it 's going to say well , nothing arrived from the sa node , so i 'm going to let off my own signal . and it 's going to keep doing this . so it 's going to go on its own rhythm now . so 2 , 3 , so all this time , the av node is on its own rhythm . and then finally , before av node is able to fire off its own fifth action potential by itself , a signal arrives from the sa node , this red arrow that i drew in . and so it 'll say , oh wait . we just got some positive ion passed through the electrical conduction system . so let 's go with it . so it 'll have a signal there . and then now , it 'll have another one here , because what happens at that point ? well , you have this guy arrives . he took 4 seconds , and he arrives right there . and then this guy is going to arrive after that . he 's going to arrives right there . so you see they start arriving . and so , once they start arriving , then you get back onto what looks like a normal rhythm . and so , it 's interesting because you basically , as a result of this long delay , have a phenomenon where for awhile , the av node is doing its own thing over here . and then after that , the sa node catches up . and then it continues on what would look like a normal sinus rhythm . and so , sometimes you 'll hear the term escape beats or escape rhythm . and so that 's what these are , these are escape beats , meaning they have escaped the normal flow of electrical conduction , which starts with the sa node . so , hope that was helpful .
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so that 's the length of a heartbeat there . and then we have the av node . i 'm just going to quickly go through this .
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are more calcium ions being added by the av node ?
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so i know we talked about different pacemakers in the body , but i thought it 'd be fun to revisit that and show you an interesting example . so let 's start out by laying out the table we 'd set up before . we talked about the heart rate in beats per minute , and we talked about the heartbeat itself -- the length of the heartbeat , and we 'd measured the heartbeat in terms of seconds . and you remember , there 's a nice little relationship between the two of these , because if the heartbeat actually gets shorter , then you can have more heart beats in a minute . and so of course , then the heart rate goes up . so that 's a relationship that explains how it is that our heart rate goes up and down . and we talked about the sa node , the av node , and the bundle of his . and we said starting with the sa node , the heart rate was somewhere between 60 and 90 . and i think i 'd chosen 90 , just because that was a nice , easy number to do math with . and we had said that the heart beat is about 0.66 seconds . so that 's the length of a heartbeat there . and then we have the av node . i 'm just going to quickly go through this . i know this is recap for you if you 've seen the other video . if you have n't , then these numbers come from basically dividing beats per minute down into seconds . and so then each beat then would be one second for the av node . and finally , we did the bundle of his . and i think i 've started trying to take a shortcut in writing bundle of his into just boh . and that looks something like this . and those underlying numbers are the numbers i 'm using to calculate the heartbeat lengths . so that 's basically what we had come up with . and we had also talked about the idea of having delays . you actually need time for the pulse to be in transit basically . and so , i 'm actually going to add a third column to our little table here . and there really is no delay here , because the sa node is where things are starting . so let me actually just keep my colors the same . and then the av node , we know that there 's a small delay , because things do move pretty quick . so we said that here , it 'd be something like 0.04 seconds . so you can see that it 's actually pretty quick getting from the sa node to the av node . but then it gets even faster as you get down to the bundle of his . it actually takes only about 0.005 seconds . so it gets really , really fast . and remember that this transit speed , this is really related to conduction velocity . so how fast is the signal getting conducted ? so we call this conduction velocity . and the relationship between conduction velocity and the action potential is the slope of phase 0 . remember , the more steep phase 0 is , the faster something is going to go from cell to cell to cell . and actually , that brings up a good point , because in the av node , there 's a huge delay built in , because the conduction is so darn slow . and so you have to actually remember that there 's this 0.1 second delay . and generally speaking , i think of 0.1 seconds as almost nothing , but when you compare it to 0.005 seconds , because that 's the transit time -- that 's how long it takes the signal to get down , we said from the av node down to our particular bundle of his cell -- then all of a sudden , this delay is looking enormous . by comparison , this looks like a really big , big number . and let me just write transit here as well . so this is time for movement . and then the delay is simply getting through the av node itself . so this is all just rehashing what we 've talked about before . and finally , just to get at least a drawing down , because i like to draw , we have our sa node here . and we have our av node here . and we have our bundle of his over here . and let me draw it half the distance , somewhere like this . and remember , this is the direction of flow . we 're basically trying to move this way . and again , this way . so let me actually jump into something slightly new . so let 's assume for a second -- this is a thought experiment -- that instead of 0.04 seconds , i 'm just going to focus on these two right now . instead of 0.04 seconds , let 's say that it took 100 times as long . for some reason , let 's say that transit time for some reason , we do n't know why , let 's say it takes 100 times longer . so this ends up being 4 seconds , right ? 0.04 times 100 is 4 seconds . so let 's say it takes about 4 seconds , for some reason , to get a signal from the sa node to the av node . well , what would that mean for us ? what would that look like exactly ? and i think you 'll start seeing some interesting lessons from this little thought experiment . so , if that was the case , if it was actually taking about 4 seconds to get from one point to another , let 's now draw out a timeline . this is a little time line , and this timeline starts at 0 seconds . and then you have , let 's say , 1 second here , 2 seconds , 3 -- i 'm just going to see how far this goes -- 4 , 5 , and let 's go to 6 . so , this is 6 . seconds and we 're going to follow what happens over 6 seconds . so let 's imagine now we keep track of our sa node up here . and we 're going to keep track of our av node down here . so at time 0 , let 's imagine that everything is beginning . and we watch our sa node , let 's start with that one first . well , at 2/3 of a second -- because that 's about how long it takes , we calculated -- we would get our first action potential , or a heart beat would go through , right ? first beat . and that would then try to make its way towards the av node . so this one is going to try to make its way towards the av node . but we know it takes 4 seconds to get there . now , what happens after that ? well , you 'd have another beat let off . the first one has n't actually made it to the av node , but the second one is already done by that point . and you 'd have a third beat that goes through by that point . and so really , we 're counting these action potentials that are going through the sa node . and they just keep going through . they 're just going to keep flowing through here . and they 're going to all just continue and basically , just what are we going to get ? a total of probably 9 , right ? we 're going to get 9 signals sent off . now , take each of them is going to take 4 seconds to get to the av node . so when will this first one get to the av node ? this very first one will get to the av node somewhere around here , because that 's 4 and 2/3 of a second . so at 4 and 2/3 of a second , this one -- let me somehow show you without making this too messy -- will make it to the av node right here . of course at that time , the sa node itself is letting out its seventh action potential , but that very first one will get there at that point . now the av node , is it going to sit around and wait for 4 and 2/3 of a second to just go by and not do anything ? no way , right ? there 's no way , because what it 's going to do is it 's going to say well , let 's wait for a signal from the sa node . and at this point , it 's going to say well , nothing arrived from the sa node , so i 'm going to let off my own signal . and it 's going to keep doing this . so it 's going to go on its own rhythm now . so 2 , 3 , so all this time , the av node is on its own rhythm . and then finally , before av node is able to fire off its own fifth action potential by itself , a signal arrives from the sa node , this red arrow that i drew in . and so it 'll say , oh wait . we just got some positive ion passed through the electrical conduction system . so let 's go with it . so it 'll have a signal there . and then now , it 'll have another one here , because what happens at that point ? well , you have this guy arrives . he took 4 seconds , and he arrives right there . and then this guy is going to arrive after that . he 's going to arrives right there . so you see they start arriving . and so , once they start arriving , then you get back onto what looks like a normal rhythm . and so , it 's interesting because you basically , as a result of this long delay , have a phenomenon where for awhile , the av node is doing its own thing over here . and then after that , the sa node catches up . and then it continues on what would look like a normal sinus rhythm . and so , sometimes you 'll hear the term escape beats or escape rhythm . and so that 's what these are , these are escape beats , meaning they have escaped the normal flow of electrical conduction , which starts with the sa node . so , hope that was helpful .
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and they 're going to all just continue and basically , just what are we going to get ? a total of probably 9 , right ? we 're going to get 9 signals sent off .
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0 in thinking about heartbeats : can monitor techs detect escape beats on an ekg and know that is what is happening ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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how was pascals triangle discovered ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here .
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is it just coincidence that the sum of the coefficients of ( a+b ) ^n = 2^n ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here .
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whats the difference between a^3 + b^3 and ( a+b ) ^3 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it .
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what if you have terms with exponents inside the brackets ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion .
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and also what do you do if there are multiple variables with exponents multiplied within the brackets ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ?
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i encountered this one reacently and became uncertain : ( 3x^3y^2z ) ^4 how would that be solved and could you use pascals triangle for this ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth .
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binomial theorum and pascal 's triangle ( -p+q ) ^5 my answer was -p^5 + 5p^4q - 10p^3q^2 + 10p^2q^3 - 5pq^4 -q^5 but the answer for the question was listed with the last term +q^5 my question is why is n't it -q^5 for the last term ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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how could you figure out numbers in larger rows of pascal 's triangle ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power .
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when you are trying to solve a problem with pascal 's triangle , you would have to memorize it or write it out again , correct ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that .
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is there a way to do the theorem with trinomials ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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is it true that we can obtain sierpinski triangle with the help of pascals triangle ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that .
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after constructing pascals triangle to a large extent , does there not become countless ways to get to every number ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here .
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how would i solve ( a+b ) ( a+b ) ( a+b ) instead of ( a+b ) ^2 ( a+b ) ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here .
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is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here .
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is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here .
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is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here .
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is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related .
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can somone show me how to expand ( x + 1 ) 1 = x + 1 please ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own .
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why does the binomial formula work ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own .
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why does the binomial formula with combinatorics work ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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this question might be stupid but , why is it that the pascal 's triangle works for figuring out leading coefficients for the binomial theorem ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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in pascal 's triangle , is term the same thing as position in that they are both 1 greater than r , in the combination ncr ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is .
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how exactly do we know that , in the video , the numbers in the bottom layer represent the coefficients of the expanded polynomial and that the layer numbers represent the exponents ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared .
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what does the numbers of ways of reaching to a point have to do anything with binomial expansion ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power .
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can we just directly add the powers like 1+3 in the 3rd level to get 4 in the fourth level in the pascal 's triangle ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared .
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why is 0 factorial equal to 1 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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why does pascal 's triangle work and how do you know when to stop going ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it .
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how is 4 choose 0 = 1 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it .
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what is the quotient of the problem 36 divided by 6 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power .
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where did you get x to the power of 6 and y to the power of 6 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power .
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0 and 8 when he adds the a to the 0th power and b to the 0th power , what would b & a be ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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how do you do the equation ( x-2 ) ^3 using pascal 's triangle ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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how would you figure out the coefficients for a bigger row like the tenth row for example without having to draw or do pascal 's triangle ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that .
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is there a similar way to calculate for binomials such as ( 2x+3y ) ^6 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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was pascal 's triangle made just for the binomial theorem ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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is it a coincidence that the 1st level of pascal 's triangle is ( 11^0 ) , the 2nd level is ( 11^1 ) and so on ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate .
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are there any other applications to using the pascal 's triangle , besides for binomial expansion ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes .
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is it possible to start pascal 's triangle with 0 instead of 1 ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top .
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what 's the difference between pascal 's triangle and the fibonacci sequence ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here .
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is there an easy way to figure out what numbers go on the next rows ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared .
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what does it mean when sal says there are two ways to get the ab term ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ?
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so if in two dimensions we can find a binomial , can we find a quadratic by adding another dimension ?
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in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it up here , at each level you 're really counting the different ways that you can get to the different nodes . so one -- and so i 'm going to set up a triangle . so if i start here there 's only one way i can get here and there 's only one way that i could get there . but now this third level -- if i were to say how many ways can i get here -- well , one way to get here , one way to get here . so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . but the way i could get here , i could go like this , or i could go like this . and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? you could go like this , you could go like this , or you could go like that . same exact logic : there 's three ways to get to this point . and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . there 's six ways to go here . three ways to get to this place , three ways to get to this place . so six ways to get to that and , if you have the time , you could figure that out . there 's three plus one -- four ways to get here . and then there 's one way to get there . and now i 'm claiming that these are the coefficients when i 'm taking something to the -- if i 'm taking something to the zeroth power . this is if i 'm taking a binomial to the first power , to the second power . obviously a binomial to the first power , the coefficients on a and b are just one and one . but when you square it , it would be a squared plus two ab plus b squared . if you take the third power , these are the coefficients -- third power . and to the fourth power , these are the coefficients . so let 's write them down . the coefficients , i 'm claiming , are going to be one , four , six , four , and one . and how do i know what the powers of a and b are going to be ? well i start a , i start this first term , at the highest power : a to the fourth . and then i go down from there . a to the fourth , a to the third , a squared , a to the first , and i guess i could write a to the zero which of course is just one . and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . this term right over here is equivalent to this term right over there . and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . well , to realize why it works let 's just go to these first levels right over here . if i just were to take a plus b to the second power . a plus b to the second power . this is going to be , we 've already seen it , this is going to be a plus b times a plus b so let me just write that down : a plus b times a plus b . so we have an a , an a . we have a b , and a b . we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . and that 's the only way . that 's the only way to get an a squared term . there 's only one way of getting an a squared term . then you 're going to have plus a times b . so -- plus a times b . and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . how many ways can you get an a squared term ? well there 's only one way . you 're multiplying this a times that a . there 's one way of getting there . now how many ways are there of getting the b squared term ? how many ways are there of getting the b squared term ? well there 's only one way . multiply this b times this b . there 's only one way of getting that . but how many ways are there of getting the ab term ? the a to the first b to the first term . well there 's two ways . you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? well there is only one way to get an a squared , there 's two ways to get an ab , and there 's only one way to get a b squared . if you set it to the third power you 'd say okay , there 's only one way to get to a to the third power . you just multiply the first a 's all together . and there is only one way to get to b to the third power . but there 's three ways to get to a squared b . and you could multiply it out , and we did it . we did it all the way back over here . there 's three ways to get a squared b . we saw that right over there . and there are three ways to get a b squared . three ways to get a b squared . and if you sum this up you have the expansion of a plus b to the third power . so hopefully you found that interesting .
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and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ?
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how does one denote the set of numbers in the range in sigma notation ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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i can not resolve this in my head : i get the math , but if at 24 years armaan is 3x as old as diya at 8 , would that mean that arman is always 3 times as old as diya ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides .
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i mean , we all experience time at a constant rate ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ?
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mary age is 2/3 that of peter , s .two years ago mary , s age was 1/2 of what peter , s age will be in 5 years time .how old is peter now ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years .
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if i am 25 years old and my little brother jonas is 5 years old , how long would it take for me to be three times jonas ' age ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old .
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what are their present ages ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ?
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benson and jackson are brother , benson age is two years less than twice jackson 's age the sum 19 years , how old each brother ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ?
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wo n't armaan and diya have the same age difference , no matter how much time passes ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years .
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how long will it take for ee to be 4 times uu 's age ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ?
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how long ago in years did ship sink ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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well , let 's think about how old arman will be in y years . how old will he be ? let me write here .
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how old is umaima now ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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well , let 's think about how old arman will be in y years . how old will he be ? let me write here .
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how old is william now ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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well , let 's think about how old arman will be in y years . how old will he be ? let me write here .
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how old is christopher now ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old .
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if the perimeter of the floor of the room is 90. find the length and the width of the room ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y .
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wait how come the equation is 8+x = 3 ( 2+y ) ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years .
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should n't sal have noticed he wrote `` armaan '' when it should be arman ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old .
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what does ''magenta '' mean ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y .
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a man is 4times as old as his son and five years ago the product of their ages is 234 find their present ages ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y .
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mrs. zamora is 36 years old and her son is 4 when will the mother be thrice as old as her son ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y .
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couldnt you of just multiplied 3 times the age of diya to equal 6 years ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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it will be 6 years before arman is 3 times as old as diya , right ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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well , let 's think about how old arman will be in y years . how old will he be ? let me write here .
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if the sum of their ages is 64 , how old is joan ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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well , let 's think about how old arman will be in y years . how old will he be ? let me write here .
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if the sum of their ages is 33 , how old is each person ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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well , let 's think about how old arman will be in y years . how old will he be ? let me write here .
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if the sum of their ages is 47 , how old is each person ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years .
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how many years will it take until christopher is only 3 times as old as gabriela ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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since arman is the one who will be 3 times older than diya , should n't arman 's age in y years be multiplied by 3 instead ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y .
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is n't arman already greater than 3 times diya 's age ?
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let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i encourage you to try to take a shot at this yourself . so let 's think about this a little bit . we 're asking how many years will it take . that 's what we do n't know . that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . let 's say y is equal to years it will take . so given that , can we now set up an equation , given this information , to figure out how many years it will take for arman to be three times as old as diya ? well , let 's think about how old arman will be in y years . how old will he be ? let me write here . in y years , arman is going to be how old ? arman is going to be -- well , he 's 18 right now -- and in y years , he 's going to be y years older . so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . so what we 're curious about , now that we know this , is how many years will it take for this quantity , for this expression , to be three times this quantity ? so we 're really curious . we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . and we 're saying that what arman 's going to be in y years is three times what diya is going to be in y years . so we 've set up our equation . now we can just solve it . so let 's take this step by step . so the left hand side -- and maybe i 'll do this in a new color , just so i do n't have to keep switching -- so on the left hand side , i still have 18 plus y . and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . and then it 's always nice to get all of our constants on one side of the equation , all of our variables on the other side of the equation . so we have a 3y over here . we have more y 's on the right hand side than the left hand side . so let 's get rid of the y 's on the left hand side . you could do it either way , but you 'd end up with negative numbers . so let 's subtract a y from each side . and we are left with , on the left hand side , 18 . and on the right hand side you have 6 plus 3 y 's . take away one of those y 's . you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . 18 minus 6 is 12 . the whole reason why we subtracted 6 from the right was to get rid of this , 6 minus 6 is 0 , so you have 12 is equal to 2y . two times the number of years it will take is 12 , and you could probably solve this in your head . but if we just want a one-coefficient year , we would divide by 2 on the right . whatever we do to one side of an equation , we have to do it on the other side . otherwise , the equation will not still be an equation . so we 're left with y is equal to 6 , or y is equal to 6 . so going back to the question , how many years will it take for arman to be three times as old as diya ? well , it 's going to take six years . now , i want you to verify this . think about it . is this actually true ? well , in six years , how old is arman going to be ? he 's going to be 18 plus 6 . we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? well , she 's going to be 2 plus 6 , which is 8 years old . and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done .
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and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y .
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a man is 4times as old as his son and five years ago the product of their ages is 234 find their present ages ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these .
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is there a limit to the amount of prepositions you can have in a sentence ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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`` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur .
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hey david , can we use until instead of by and vice versa as both of them indicate a precise ending ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . ''
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do i can use 'on ' only at specific day ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english .
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is there any difference between till and until ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock .
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how would one distinguish use of since as a conjunction versus the standard conjunction of because ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . ''
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what is the difference between 'by ' and 'until ' with reference to time ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything .
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and respectively , would n't those usages of `` after '' and `` before '' both be considered a conjunction instead of a preposition ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise .
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but how can you take a garbage out , if you have n't leaved the house ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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`` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three .
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i 'm not a native speaker , so it 's weird to me ... but why does the word `` had '' is in the sentence `` this place had better be clean by 3 p.m. '' ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool .
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if the prepositions of time were used to denote space , then what spatial relation did for denote ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool .
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can you use a time preposition as a space preposition as well ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail .
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why in in the video the bat drawing suddenly finishes the legs ?
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards wear multiple , simultaneous hats . if you want to wear a second hat on top of your other time wizard hat , that 's the silliest thing i 've ever said , then you will have to learn the prepositions of time . now here 's something weird and cool about prepositions of time is that once upon a time all of these were physical prepositions . like `` before '' and `` after '' just used to mean behind and in front of . they later took on this additional connotation of time just over the course of english-speaking history . people took this literal meaning of `` before '' and `` after '' and made it representational . this moment occurred in front of this one in time , but behind this other one in time . it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . `` after '' and `` before , '' as we 've established , these are time relationships that refer to something happening after . so when something is completed , say , `` the bats come out after the sun goes down . '' and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . when we 're talking about `` at '' we 're talking about a single moment in time . we could say , `` the vampire wakes at 10 p.m. '' there he is emerging from his coffin . there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . it 's got a little stubby tail . doop . ah , i really want a bat clock now . anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . that would be nice , but it 's only relevant to you that the cutoff time is at three o'clock . so the end is precise . that 's the connotation there . but the beginning is not . `` for '' denotes duration . how long something has been going on . so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . `` in '' denotes a bounded duration . so it 's something that lasts for a specific amount of time , like a limited period . okay , so let 's just say bounded duration . so that covers usage like , `` in march '' or `` in the middle ages . '' both of those things are like set periods . march has a beginning and an end . the middle ages have a beginning and an end . it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' mind you not everybody eats hot dogs or likes fireworks , so i said many not all . `` since '' is kind of like `` by '' except it 's more about the precision of the start point rather than the end point . so , precise beginning . `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . all right . so there 's a precise ending there . you have `` until midnight , '' and then you ca n't rescue the ambassador , break the curse or save any princes . but what you can do is learn anything . these are some of the most essential time prepositions . david out .
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hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english .
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is `` during '' a preposition ?
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we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends on the hybridization state of this carbon . so let 's look at some examples here . so , if we look at an example where the carbon is sp-hybridized , so we know this is an sp-hybridized carbon because we have a triple bond here , so we 're talking about a carbon-hydrogen bond , where the carbon is sp-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3300 wavenumbers . if we look at this next example here , so now we have a carbon that has a double bond to it , so it must be sp2-hybridized , so we 're talking about a carbon-hydrogen bond , where the carbon is sp2-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3100 wavenumbers . and then finally , if we look at a situation where we have only single bonds to this carbon , we 're talking about an sp3-hybridized carbon here , so we 're talking about a carbon-hydrogen bond where the carbon is sp3-hybridized . the signal for this carbon-hydrogen bond stretch shows up about 2900 wavenumbers . and so , how do we explain these different wavenumbers , because they 're all carbon-hydrogen bonds ? well , we need to think about the hybridizations . so let 's do that . so if we look at the sp-hybridized carbon , remember , that means that this carbon has two sp-hybrid orbitals . and an sp-hybrid orbital has the most s character out of all these orbitals we 've discussed here . so , actually 50 % s character , if you remember that from the videos on hybridizations . so 50 % s character for an sp-hybridized orbital . for an sp2-hybridized orbital , it 's about 33 % s character . and finally for an sp3-hybridized orbital , it 's about 25 % s character . and so going back to the sp-hybridized carbon , so the sp-hybrid orbital is 50 % s character . that means -- remember what that means . the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . and if this is the shortest bond , it must be the strongest bond out of these three that we 're talking about . so this carbon-hydrogen bond , where the carbon is sp-hybridized , is stronger that the carbon-hydrogen bond where the carbon is sp2-hybridized . this bond , though , has more s character than this one , so this bond is stronger than this one . so this order of bond strength explains the wavenumbers , because if you remember from the previous video , the bond strength affects the force constant , or the spring constant k , so as you increase in bond strength , you increase k , and we saw that increasing k increases the frequency , or the wavenumber . so this increases the frequency of bond vibrations , or increases the wavenumber where you would find the signal on your ir spectrum . and so since this is the strongest bond , this is the highest value for the wavenumber , so we 're going to find this signal more to the left on our ir spectrum when we 're looking at it . alright , now that we understand this idea , so the hybridization , we can look at some ir spectra for hydrocarbons , and we can analyze those . let 's do that . first , let 's compare alkanes and alkynes . let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . the second one is for 1-octyne , so a triple bond in this molecule . let 's compare these two , so you think about the differences . alright , one of the things that 's sometimes helpful to do is to draw a line around 3000 . so let me draw a line around 3000 . i 'm going to try and draw it for both here too . so i 'm going to draw a line around 3000 for both , so we can compare these two spectra . alright , we know that a carbon-hydrogen -- let me go and write this in here , a carbon-hydrogen bond where the carbon is sp3-hybridized , so that signal for that stretch shows up under 3000 . so that 's why it 's helpful to draw a line around 3000 . so that 's what we 're talking about when we 're talking about this complicated-looking thing in here . so it 's not really worth your time to analyze this in great detail , and of course my drawing of it is n't perfect to being with , but think about under 3000 , that 's where you expect to find your signal for your carbon-hydrogen bond , we 're talking about an sp3-hybridized carbon . and those are the only types of carbons that we have in decane . so , if we think about the diagnostic region versus the fingerprint region , so if i draw a line here to separate those two regions , in the diagnostic region , all we have is this . so all we 're thinking about is carbon-hydrogen where the carbon is sp3-hybridized . so very simple spectrum to analyze . we move on to 1-octyne , so now we 're looking at this one down here . so we see that same kind of thing , because obviously we have carbon-hydrogen sp3-hybridized also in this molecule . and so this is n't going to really help you too much when you 're analyzing the spectra , but it 's useful to know what you 're looking at , drawing a line at 3000 , and thinking about that 's what that represents . so once you draw a line at 3000 , it allows you to see some differences . so for example , this signal right here , if we drop down , it 's pretty close to 3300 , so this will be 3100 , 3200 , so 3300 . so approximately 3300 wavenumbers . and we know what that signal represents . we can go back up to here , we can look at about 3300 is where we would expect to see this carbon-hydrogen bond stretch where that carbon is sp-hybridized . so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . we also have a carbon right here , so that gives you your eight carbons . and so this bond , let me go and highlight it here , so this bond right here , this carbon-hydrogen bond , where this carbon is sp-hybridized , that 's this signal on our spectrum . so once again , it 's useful for analyzing here . alright , we have something else that shows up in the diagnostic region for this alkyne , and it 's this signal right here . so if we drop down , what 's the wavenumber where this signal appears ? the wavenumber is about 2100 , so approximately 2100 , maybe a little bit higher than that . and that 's the carbon triple bond stretch , so that 's the carbon-carbon triple bond stretch that we talked about in an earlier video . so that 's approximately the triple bond region when you 're looking at your spectrum . and of course , obviously we have one . this is an alkyne here . and so hopefully , this just shows you the differences , and once again , your fingerprint region over here is unique for each of these molecules here . so this shows you the differences and helps you to think about how to analyze your ir spectrum . let 's look at two more . actually , let 's look at one more here , and let 's compare these two . so now we have a spectrum for an alkene . so here we have 1-hexene , and let 's see if we can analyze this one . so we 're going to do the same thing , we 're going to draw a line around 1500 , so draw a line around 1500 , we 're going to draw a line around 3000 , and let 's analyze this one . so we know what this one is talking about , we know it 's talking about a carbon-hydrogen , where the carbon is sp3-hybridized . but what 's this signal right here ? we drop down , and that 's pretty close to 3100 . so that signal is approximately 3100 . and so , we know what that must represent . that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here . so this signal shows up around 3100 . so i draw a line up here , and we saw this signal at 3300 . so it 's useful to think about , it helps you distinguish on your ir spectra , if you draw a line in there , and think about where the wavenumber is . at what wavenumber does this signal appear ? we also have something else showing up in this one , let me go ahead and draw this one in here . what is this ? what is this guy right here ? that looks like a pretty obvious signal . so we can drop down here , and check out approximately where does that show up ? what 's that wavenumber ? well this is 1500 , this is 1600 , this is 1700 , so that 's a little bit -- that 's pretty much in between , it 's approximately 1650 . and in an earlier video , we said that was in the double bond region . so that 's the carbon-carbon double bond stretch signal , right in here . and obviously , there 's a double bond in our molecule . again , comparing these two , remember , we talked about the fact that a triple bond vibrates faster than a double bond . and so , the signal is different . the triple bond vibrates faster , so it has a higher wavenumber , the double bond does n't vibrate as fast , so it has a lower wavenumber . so these are important things to think about . finally , let 's compare this alkene to an arene here . so let 's look at toluene right here . alright , so if we do the same thing , if we draw a line around 3000 , so somewhere around 3000 . and we know this is below 3000 , so we know that this must be talking about carbon-hydrogen , where the carbon is sp3-hybridized . that 's a carbon-hydrogen bond stretch where the carbon is sp3-hybridized . well this carbon right here on toluene , so this is toluene , that carbon is sp3-hybridized , so that makes sense . we have this one little peak here , this one little signal that 's a little bit higher than 3000 , and so it 's pretty close to 3100 . and we know that 's approximately where we would find a carbon-hydrogen bond stretch where the carbon is sp2-hybridized . so somewhere around 3100 . and so , this makes this -- at first you might think , `` oh , well how do we tell this apart ? '' so this is very similar to this situation . so this looks very similar to this . and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . i 'm going to draw a line down here , and let 's try to connect the 1600s right here like that . and let 's think about what happened . so here 's the signal for the carbon-carbon double bond here , and when you 're talking about an aromatic double bond stretch , so carbon-carbon aromatic , that usually shows up lower than 1600 . so here it looks like we have two signals , so it depends on what kind of compound you 're dealing with . but we can see that this usually shows up lower . so this is somewhere usually around 1600 to 1450 . but we 're talking about the carbon-carbon double bond stretching here . and there 's some other subtle things that can clue you in , like this right here . so we do n't really see that on this one , on this spectrum . and then , these down here , we 're missing those here too . it would be way too much to get into in this video , to talk about what these mean , and that 's a little bit more than what we 've talked about so far , so that 'll have to be a different video . but there are subtle differences , and the easiest one to think about is to think about the fact that this aromatic carbon-carbon double bond shows up at a lower wavenumber than the one that we talked about right here . so just look at the sheer multitude of signals and sometimes that clues you into the fact that you 're dealing with this benzene ring here for toluene . so that sums up just a quick intro to looking at ir spectra for hydrocarbons .
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and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one .
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in which region does the c-c bond appear ?
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we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends on the hybridization state of this carbon . so let 's look at some examples here . so , if we look at an example where the carbon is sp-hybridized , so we know this is an sp-hybridized carbon because we have a triple bond here , so we 're talking about a carbon-hydrogen bond , where the carbon is sp-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3300 wavenumbers . if we look at this next example here , so now we have a carbon that has a double bond to it , so it must be sp2-hybridized , so we 're talking about a carbon-hydrogen bond , where the carbon is sp2-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3100 wavenumbers . and then finally , if we look at a situation where we have only single bonds to this carbon , we 're talking about an sp3-hybridized carbon here , so we 're talking about a carbon-hydrogen bond where the carbon is sp3-hybridized . the signal for this carbon-hydrogen bond stretch shows up about 2900 wavenumbers . and so , how do we explain these different wavenumbers , because they 're all carbon-hydrogen bonds ? well , we need to think about the hybridizations . so let 's do that . so if we look at the sp-hybridized carbon , remember , that means that this carbon has two sp-hybrid orbitals . and an sp-hybrid orbital has the most s character out of all these orbitals we 've discussed here . so , actually 50 % s character , if you remember that from the videos on hybridizations . so 50 % s character for an sp-hybridized orbital . for an sp2-hybridized orbital , it 's about 33 % s character . and finally for an sp3-hybridized orbital , it 's about 25 % s character . and so going back to the sp-hybridized carbon , so the sp-hybrid orbital is 50 % s character . that means -- remember what that means . the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . and if this is the shortest bond , it must be the strongest bond out of these three that we 're talking about . so this carbon-hydrogen bond , where the carbon is sp-hybridized , is stronger that the carbon-hydrogen bond where the carbon is sp2-hybridized . this bond , though , has more s character than this one , so this bond is stronger than this one . so this order of bond strength explains the wavenumbers , because if you remember from the previous video , the bond strength affects the force constant , or the spring constant k , so as you increase in bond strength , you increase k , and we saw that increasing k increases the frequency , or the wavenumber . so this increases the frequency of bond vibrations , or increases the wavenumber where you would find the signal on your ir spectrum . and so since this is the strongest bond , this is the highest value for the wavenumber , so we 're going to find this signal more to the left on our ir spectrum when we 're looking at it . alright , now that we understand this idea , so the hybridization , we can look at some ir spectra for hydrocarbons , and we can analyze those . let 's do that . first , let 's compare alkanes and alkynes . let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . the second one is for 1-octyne , so a triple bond in this molecule . let 's compare these two , so you think about the differences . alright , one of the things that 's sometimes helpful to do is to draw a line around 3000 . so let me draw a line around 3000 . i 'm going to try and draw it for both here too . so i 'm going to draw a line around 3000 for both , so we can compare these two spectra . alright , we know that a carbon-hydrogen -- let me go and write this in here , a carbon-hydrogen bond where the carbon is sp3-hybridized , so that signal for that stretch shows up under 3000 . so that 's why it 's helpful to draw a line around 3000 . so that 's what we 're talking about when we 're talking about this complicated-looking thing in here . so it 's not really worth your time to analyze this in great detail , and of course my drawing of it is n't perfect to being with , but think about under 3000 , that 's where you expect to find your signal for your carbon-hydrogen bond , we 're talking about an sp3-hybridized carbon . and those are the only types of carbons that we have in decane . so , if we think about the diagnostic region versus the fingerprint region , so if i draw a line here to separate those two regions , in the diagnostic region , all we have is this . so all we 're thinking about is carbon-hydrogen where the carbon is sp3-hybridized . so very simple spectrum to analyze . we move on to 1-octyne , so now we 're looking at this one down here . so we see that same kind of thing , because obviously we have carbon-hydrogen sp3-hybridized also in this molecule . and so this is n't going to really help you too much when you 're analyzing the spectra , but it 's useful to know what you 're looking at , drawing a line at 3000 , and thinking about that 's what that represents . so once you draw a line at 3000 , it allows you to see some differences . so for example , this signal right here , if we drop down , it 's pretty close to 3300 , so this will be 3100 , 3200 , so 3300 . so approximately 3300 wavenumbers . and we know what that signal represents . we can go back up to here , we can look at about 3300 is where we would expect to see this carbon-hydrogen bond stretch where that carbon is sp-hybridized . so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . we also have a carbon right here , so that gives you your eight carbons . and so this bond , let me go and highlight it here , so this bond right here , this carbon-hydrogen bond , where this carbon is sp-hybridized , that 's this signal on our spectrum . so once again , it 's useful for analyzing here . alright , we have something else that shows up in the diagnostic region for this alkyne , and it 's this signal right here . so if we drop down , what 's the wavenumber where this signal appears ? the wavenumber is about 2100 , so approximately 2100 , maybe a little bit higher than that . and that 's the carbon triple bond stretch , so that 's the carbon-carbon triple bond stretch that we talked about in an earlier video . so that 's approximately the triple bond region when you 're looking at your spectrum . and of course , obviously we have one . this is an alkyne here . and so hopefully , this just shows you the differences , and once again , your fingerprint region over here is unique for each of these molecules here . so this shows you the differences and helps you to think about how to analyze your ir spectrum . let 's look at two more . actually , let 's look at one more here , and let 's compare these two . so now we have a spectrum for an alkene . so here we have 1-hexene , and let 's see if we can analyze this one . so we 're going to do the same thing , we 're going to draw a line around 1500 , so draw a line around 1500 , we 're going to draw a line around 3000 , and let 's analyze this one . so we know what this one is talking about , we know it 's talking about a carbon-hydrogen , where the carbon is sp3-hybridized . but what 's this signal right here ? we drop down , and that 's pretty close to 3100 . so that signal is approximately 3100 . and so , we know what that must represent . that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here . so this signal shows up around 3100 . so i draw a line up here , and we saw this signal at 3300 . so it 's useful to think about , it helps you distinguish on your ir spectra , if you draw a line in there , and think about where the wavenumber is . at what wavenumber does this signal appear ? we also have something else showing up in this one , let me go ahead and draw this one in here . what is this ? what is this guy right here ? that looks like a pretty obvious signal . so we can drop down here , and check out approximately where does that show up ? what 's that wavenumber ? well this is 1500 , this is 1600 , this is 1700 , so that 's a little bit -- that 's pretty much in between , it 's approximately 1650 . and in an earlier video , we said that was in the double bond region . so that 's the carbon-carbon double bond stretch signal , right in here . and obviously , there 's a double bond in our molecule . again , comparing these two , remember , we talked about the fact that a triple bond vibrates faster than a double bond . and so , the signal is different . the triple bond vibrates faster , so it has a higher wavenumber , the double bond does n't vibrate as fast , so it has a lower wavenumber . so these are important things to think about . finally , let 's compare this alkene to an arene here . so let 's look at toluene right here . alright , so if we do the same thing , if we draw a line around 3000 , so somewhere around 3000 . and we know this is below 3000 , so we know that this must be talking about carbon-hydrogen , where the carbon is sp3-hybridized . that 's a carbon-hydrogen bond stretch where the carbon is sp3-hybridized . well this carbon right here on toluene , so this is toluene , that carbon is sp3-hybridized , so that makes sense . we have this one little peak here , this one little signal that 's a little bit higher than 3000 , and so it 's pretty close to 3100 . and we know that 's approximately where we would find a carbon-hydrogen bond stretch where the carbon is sp2-hybridized . so somewhere around 3100 . and so , this makes this -- at first you might think , `` oh , well how do we tell this apart ? '' so this is very similar to this situation . so this looks very similar to this . and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . i 'm going to draw a line down here , and let 's try to connect the 1600s right here like that . and let 's think about what happened . so here 's the signal for the carbon-carbon double bond here , and when you 're talking about an aromatic double bond stretch , so carbon-carbon aromatic , that usually shows up lower than 1600 . so here it looks like we have two signals , so it depends on what kind of compound you 're dealing with . but we can see that this usually shows up lower . so this is somewhere usually around 1600 to 1450 . but we 're talking about the carbon-carbon double bond stretching here . and there 's some other subtle things that can clue you in , like this right here . so we do n't really see that on this one , on this spectrum . and then , these down here , we 're missing those here too . it would be way too much to get into in this video , to talk about what these mean , and that 's a little bit more than what we 've talked about so far , so that 'll have to be a different video . but there are subtle differences , and the easiest one to think about is to think about the fact that this aromatic carbon-carbon double bond shows up at a lower wavenumber than the one that we talked about right here . so just look at the sheer multitude of signals and sometimes that clues you into the fact that you 're dealing with this benzene ring here for toluene . so that sums up just a quick intro to looking at ir spectra for hydrocarbons .
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that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here .
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can one figure out the number of certain functional group from just analyzing the ir diagram ?
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we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends on the hybridization state of this carbon . so let 's look at some examples here . so , if we look at an example where the carbon is sp-hybridized , so we know this is an sp-hybridized carbon because we have a triple bond here , so we 're talking about a carbon-hydrogen bond , where the carbon is sp-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3300 wavenumbers . if we look at this next example here , so now we have a carbon that has a double bond to it , so it must be sp2-hybridized , so we 're talking about a carbon-hydrogen bond , where the carbon is sp2-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3100 wavenumbers . and then finally , if we look at a situation where we have only single bonds to this carbon , we 're talking about an sp3-hybridized carbon here , so we 're talking about a carbon-hydrogen bond where the carbon is sp3-hybridized . the signal for this carbon-hydrogen bond stretch shows up about 2900 wavenumbers . and so , how do we explain these different wavenumbers , because they 're all carbon-hydrogen bonds ? well , we need to think about the hybridizations . so let 's do that . so if we look at the sp-hybridized carbon , remember , that means that this carbon has two sp-hybrid orbitals . and an sp-hybrid orbital has the most s character out of all these orbitals we 've discussed here . so , actually 50 % s character , if you remember that from the videos on hybridizations . so 50 % s character for an sp-hybridized orbital . for an sp2-hybridized orbital , it 's about 33 % s character . and finally for an sp3-hybridized orbital , it 's about 25 % s character . and so going back to the sp-hybridized carbon , so the sp-hybrid orbital is 50 % s character . that means -- remember what that means . the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . and if this is the shortest bond , it must be the strongest bond out of these three that we 're talking about . so this carbon-hydrogen bond , where the carbon is sp-hybridized , is stronger that the carbon-hydrogen bond where the carbon is sp2-hybridized . this bond , though , has more s character than this one , so this bond is stronger than this one . so this order of bond strength explains the wavenumbers , because if you remember from the previous video , the bond strength affects the force constant , or the spring constant k , so as you increase in bond strength , you increase k , and we saw that increasing k increases the frequency , or the wavenumber . so this increases the frequency of bond vibrations , or increases the wavenumber where you would find the signal on your ir spectrum . and so since this is the strongest bond , this is the highest value for the wavenumber , so we 're going to find this signal more to the left on our ir spectrum when we 're looking at it . alright , now that we understand this idea , so the hybridization , we can look at some ir spectra for hydrocarbons , and we can analyze those . let 's do that . first , let 's compare alkanes and alkynes . let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . the second one is for 1-octyne , so a triple bond in this molecule . let 's compare these two , so you think about the differences . alright , one of the things that 's sometimes helpful to do is to draw a line around 3000 . so let me draw a line around 3000 . i 'm going to try and draw it for both here too . so i 'm going to draw a line around 3000 for both , so we can compare these two spectra . alright , we know that a carbon-hydrogen -- let me go and write this in here , a carbon-hydrogen bond where the carbon is sp3-hybridized , so that signal for that stretch shows up under 3000 . so that 's why it 's helpful to draw a line around 3000 . so that 's what we 're talking about when we 're talking about this complicated-looking thing in here . so it 's not really worth your time to analyze this in great detail , and of course my drawing of it is n't perfect to being with , but think about under 3000 , that 's where you expect to find your signal for your carbon-hydrogen bond , we 're talking about an sp3-hybridized carbon . and those are the only types of carbons that we have in decane . so , if we think about the diagnostic region versus the fingerprint region , so if i draw a line here to separate those two regions , in the diagnostic region , all we have is this . so all we 're thinking about is carbon-hydrogen where the carbon is sp3-hybridized . so very simple spectrum to analyze . we move on to 1-octyne , so now we 're looking at this one down here . so we see that same kind of thing , because obviously we have carbon-hydrogen sp3-hybridized also in this molecule . and so this is n't going to really help you too much when you 're analyzing the spectra , but it 's useful to know what you 're looking at , drawing a line at 3000 , and thinking about that 's what that represents . so once you draw a line at 3000 , it allows you to see some differences . so for example , this signal right here , if we drop down , it 's pretty close to 3300 , so this will be 3100 , 3200 , so 3300 . so approximately 3300 wavenumbers . and we know what that signal represents . we can go back up to here , we can look at about 3300 is where we would expect to see this carbon-hydrogen bond stretch where that carbon is sp-hybridized . so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . we also have a carbon right here , so that gives you your eight carbons . and so this bond , let me go and highlight it here , so this bond right here , this carbon-hydrogen bond , where this carbon is sp-hybridized , that 's this signal on our spectrum . so once again , it 's useful for analyzing here . alright , we have something else that shows up in the diagnostic region for this alkyne , and it 's this signal right here . so if we drop down , what 's the wavenumber where this signal appears ? the wavenumber is about 2100 , so approximately 2100 , maybe a little bit higher than that . and that 's the carbon triple bond stretch , so that 's the carbon-carbon triple bond stretch that we talked about in an earlier video . so that 's approximately the triple bond region when you 're looking at your spectrum . and of course , obviously we have one . this is an alkyne here . and so hopefully , this just shows you the differences , and once again , your fingerprint region over here is unique for each of these molecules here . so this shows you the differences and helps you to think about how to analyze your ir spectrum . let 's look at two more . actually , let 's look at one more here , and let 's compare these two . so now we have a spectrum for an alkene . so here we have 1-hexene , and let 's see if we can analyze this one . so we 're going to do the same thing , we 're going to draw a line around 1500 , so draw a line around 1500 , we 're going to draw a line around 3000 , and let 's analyze this one . so we know what this one is talking about , we know it 's talking about a carbon-hydrogen , where the carbon is sp3-hybridized . but what 's this signal right here ? we drop down , and that 's pretty close to 3100 . so that signal is approximately 3100 . and so , we know what that must represent . that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here . so this signal shows up around 3100 . so i draw a line up here , and we saw this signal at 3300 . so it 's useful to think about , it helps you distinguish on your ir spectra , if you draw a line in there , and think about where the wavenumber is . at what wavenumber does this signal appear ? we also have something else showing up in this one , let me go ahead and draw this one in here . what is this ? what is this guy right here ? that looks like a pretty obvious signal . so we can drop down here , and check out approximately where does that show up ? what 's that wavenumber ? well this is 1500 , this is 1600 , this is 1700 , so that 's a little bit -- that 's pretty much in between , it 's approximately 1650 . and in an earlier video , we said that was in the double bond region . so that 's the carbon-carbon double bond stretch signal , right in here . and obviously , there 's a double bond in our molecule . again , comparing these two , remember , we talked about the fact that a triple bond vibrates faster than a double bond . and so , the signal is different . the triple bond vibrates faster , so it has a higher wavenumber , the double bond does n't vibrate as fast , so it has a lower wavenumber . so these are important things to think about . finally , let 's compare this alkene to an arene here . so let 's look at toluene right here . alright , so if we do the same thing , if we draw a line around 3000 , so somewhere around 3000 . and we know this is below 3000 , so we know that this must be talking about carbon-hydrogen , where the carbon is sp3-hybridized . that 's a carbon-hydrogen bond stretch where the carbon is sp3-hybridized . well this carbon right here on toluene , so this is toluene , that carbon is sp3-hybridized , so that makes sense . we have this one little peak here , this one little signal that 's a little bit higher than 3000 , and so it 's pretty close to 3100 . and we know that 's approximately where we would find a carbon-hydrogen bond stretch where the carbon is sp2-hybridized . so somewhere around 3100 . and so , this makes this -- at first you might think , `` oh , well how do we tell this apart ? '' so this is very similar to this situation . so this looks very similar to this . and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . i 'm going to draw a line down here , and let 's try to connect the 1600s right here like that . and let 's think about what happened . so here 's the signal for the carbon-carbon double bond here , and when you 're talking about an aromatic double bond stretch , so carbon-carbon aromatic , that usually shows up lower than 1600 . so here it looks like we have two signals , so it depends on what kind of compound you 're dealing with . but we can see that this usually shows up lower . so this is somewhere usually around 1600 to 1450 . but we 're talking about the carbon-carbon double bond stretching here . and there 's some other subtle things that can clue you in , like this right here . so we do n't really see that on this one , on this spectrum . and then , these down here , we 're missing those here too . it would be way too much to get into in this video , to talk about what these mean , and that 's a little bit more than what we 've talked about so far , so that 'll have to be a different video . but there are subtle differences , and the easiest one to think about is to think about the fact that this aromatic carbon-carbon double bond shows up at a lower wavenumber than the one that we talked about right here . so just look at the sheer multitude of signals and sometimes that clues you into the fact that you 're dealing with this benzene ring here for toluene . so that sums up just a quick intro to looking at ir spectra for hydrocarbons .
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let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there .
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for example , would two c ( sp2 ) -h 's ina molecule generate somewhat different ir diagram from a molecule that has just one c ( sp2 ) -h ?
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we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends on the hybridization state of this carbon . so let 's look at some examples here . so , if we look at an example where the carbon is sp-hybridized , so we know this is an sp-hybridized carbon because we have a triple bond here , so we 're talking about a carbon-hydrogen bond , where the carbon is sp-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3300 wavenumbers . if we look at this next example here , so now we have a carbon that has a double bond to it , so it must be sp2-hybridized , so we 're talking about a carbon-hydrogen bond , where the carbon is sp2-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3100 wavenumbers . and then finally , if we look at a situation where we have only single bonds to this carbon , we 're talking about an sp3-hybridized carbon here , so we 're talking about a carbon-hydrogen bond where the carbon is sp3-hybridized . the signal for this carbon-hydrogen bond stretch shows up about 2900 wavenumbers . and so , how do we explain these different wavenumbers , because they 're all carbon-hydrogen bonds ? well , we need to think about the hybridizations . so let 's do that . so if we look at the sp-hybridized carbon , remember , that means that this carbon has two sp-hybrid orbitals . and an sp-hybrid orbital has the most s character out of all these orbitals we 've discussed here . so , actually 50 % s character , if you remember that from the videos on hybridizations . so 50 % s character for an sp-hybridized orbital . for an sp2-hybridized orbital , it 's about 33 % s character . and finally for an sp3-hybridized orbital , it 's about 25 % s character . and so going back to the sp-hybridized carbon , so the sp-hybrid orbital is 50 % s character . that means -- remember what that means . the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . and if this is the shortest bond , it must be the strongest bond out of these three that we 're talking about . so this carbon-hydrogen bond , where the carbon is sp-hybridized , is stronger that the carbon-hydrogen bond where the carbon is sp2-hybridized . this bond , though , has more s character than this one , so this bond is stronger than this one . so this order of bond strength explains the wavenumbers , because if you remember from the previous video , the bond strength affects the force constant , or the spring constant k , so as you increase in bond strength , you increase k , and we saw that increasing k increases the frequency , or the wavenumber . so this increases the frequency of bond vibrations , or increases the wavenumber where you would find the signal on your ir spectrum . and so since this is the strongest bond , this is the highest value for the wavenumber , so we 're going to find this signal more to the left on our ir spectrum when we 're looking at it . alright , now that we understand this idea , so the hybridization , we can look at some ir spectra for hydrocarbons , and we can analyze those . let 's do that . first , let 's compare alkanes and alkynes . let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . the second one is for 1-octyne , so a triple bond in this molecule . let 's compare these two , so you think about the differences . alright , one of the things that 's sometimes helpful to do is to draw a line around 3000 . so let me draw a line around 3000 . i 'm going to try and draw it for both here too . so i 'm going to draw a line around 3000 for both , so we can compare these two spectra . alright , we know that a carbon-hydrogen -- let me go and write this in here , a carbon-hydrogen bond where the carbon is sp3-hybridized , so that signal for that stretch shows up under 3000 . so that 's why it 's helpful to draw a line around 3000 . so that 's what we 're talking about when we 're talking about this complicated-looking thing in here . so it 's not really worth your time to analyze this in great detail , and of course my drawing of it is n't perfect to being with , but think about under 3000 , that 's where you expect to find your signal for your carbon-hydrogen bond , we 're talking about an sp3-hybridized carbon . and those are the only types of carbons that we have in decane . so , if we think about the diagnostic region versus the fingerprint region , so if i draw a line here to separate those two regions , in the diagnostic region , all we have is this . so all we 're thinking about is carbon-hydrogen where the carbon is sp3-hybridized . so very simple spectrum to analyze . we move on to 1-octyne , so now we 're looking at this one down here . so we see that same kind of thing , because obviously we have carbon-hydrogen sp3-hybridized also in this molecule . and so this is n't going to really help you too much when you 're analyzing the spectra , but it 's useful to know what you 're looking at , drawing a line at 3000 , and thinking about that 's what that represents . so once you draw a line at 3000 , it allows you to see some differences . so for example , this signal right here , if we drop down , it 's pretty close to 3300 , so this will be 3100 , 3200 , so 3300 . so approximately 3300 wavenumbers . and we know what that signal represents . we can go back up to here , we can look at about 3300 is where we would expect to see this carbon-hydrogen bond stretch where that carbon is sp-hybridized . so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . we also have a carbon right here , so that gives you your eight carbons . and so this bond , let me go and highlight it here , so this bond right here , this carbon-hydrogen bond , where this carbon is sp-hybridized , that 's this signal on our spectrum . so once again , it 's useful for analyzing here . alright , we have something else that shows up in the diagnostic region for this alkyne , and it 's this signal right here . so if we drop down , what 's the wavenumber where this signal appears ? the wavenumber is about 2100 , so approximately 2100 , maybe a little bit higher than that . and that 's the carbon triple bond stretch , so that 's the carbon-carbon triple bond stretch that we talked about in an earlier video . so that 's approximately the triple bond region when you 're looking at your spectrum . and of course , obviously we have one . this is an alkyne here . and so hopefully , this just shows you the differences , and once again , your fingerprint region over here is unique for each of these molecules here . so this shows you the differences and helps you to think about how to analyze your ir spectrum . let 's look at two more . actually , let 's look at one more here , and let 's compare these two . so now we have a spectrum for an alkene . so here we have 1-hexene , and let 's see if we can analyze this one . so we 're going to do the same thing , we 're going to draw a line around 1500 , so draw a line around 1500 , we 're going to draw a line around 3000 , and let 's analyze this one . so we know what this one is talking about , we know it 's talking about a carbon-hydrogen , where the carbon is sp3-hybridized . but what 's this signal right here ? we drop down , and that 's pretty close to 3100 . so that signal is approximately 3100 . and so , we know what that must represent . that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here . so this signal shows up around 3100 . so i draw a line up here , and we saw this signal at 3300 . so it 's useful to think about , it helps you distinguish on your ir spectra , if you draw a line in there , and think about where the wavenumber is . at what wavenumber does this signal appear ? we also have something else showing up in this one , let me go ahead and draw this one in here . what is this ? what is this guy right here ? that looks like a pretty obvious signal . so we can drop down here , and check out approximately where does that show up ? what 's that wavenumber ? well this is 1500 , this is 1600 , this is 1700 , so that 's a little bit -- that 's pretty much in between , it 's approximately 1650 . and in an earlier video , we said that was in the double bond region . so that 's the carbon-carbon double bond stretch signal , right in here . and obviously , there 's a double bond in our molecule . again , comparing these two , remember , we talked about the fact that a triple bond vibrates faster than a double bond . and so , the signal is different . the triple bond vibrates faster , so it has a higher wavenumber , the double bond does n't vibrate as fast , so it has a lower wavenumber . so these are important things to think about . finally , let 's compare this alkene to an arene here . so let 's look at toluene right here . alright , so if we do the same thing , if we draw a line around 3000 , so somewhere around 3000 . and we know this is below 3000 , so we know that this must be talking about carbon-hydrogen , where the carbon is sp3-hybridized . that 's a carbon-hydrogen bond stretch where the carbon is sp3-hybridized . well this carbon right here on toluene , so this is toluene , that carbon is sp3-hybridized , so that makes sense . we have this one little peak here , this one little signal that 's a little bit higher than 3000 , and so it 's pretty close to 3100 . and we know that 's approximately where we would find a carbon-hydrogen bond stretch where the carbon is sp2-hybridized . so somewhere around 3100 . and so , this makes this -- at first you might think , `` oh , well how do we tell this apart ? '' so this is very similar to this situation . so this looks very similar to this . and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . i 'm going to draw a line down here , and let 's try to connect the 1600s right here like that . and let 's think about what happened . so here 's the signal for the carbon-carbon double bond here , and when you 're talking about an aromatic double bond stretch , so carbon-carbon aromatic , that usually shows up lower than 1600 . so here it looks like we have two signals , so it depends on what kind of compound you 're dealing with . but we can see that this usually shows up lower . so this is somewhere usually around 1600 to 1450 . but we 're talking about the carbon-carbon double bond stretching here . and there 's some other subtle things that can clue you in , like this right here . so we do n't really see that on this one , on this spectrum . and then , these down here , we 're missing those here too . it would be way too much to get into in this video , to talk about what these mean , and that 's a little bit more than what we 've talked about so far , so that 'll have to be a different video . but there are subtle differences , and the easiest one to think about is to think about the fact that this aromatic carbon-carbon double bond shows up at a lower wavenumber than the one that we talked about right here . so just look at the sheer multitude of signals and sometimes that clues you into the fact that you 're dealing with this benzene ring here for toluene . so that sums up just a quick intro to looking at ir spectra for hydrocarbons .
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so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here .
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would the peak broader or deeper for the molecule having two c ( sp2 ) -h ?
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we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends on the hybridization state of this carbon . so let 's look at some examples here . so , if we look at an example where the carbon is sp-hybridized , so we know this is an sp-hybridized carbon because we have a triple bond here , so we 're talking about a carbon-hydrogen bond , where the carbon is sp-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3300 wavenumbers . if we look at this next example here , so now we have a carbon that has a double bond to it , so it must be sp2-hybridized , so we 're talking about a carbon-hydrogen bond , where the carbon is sp2-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3100 wavenumbers . and then finally , if we look at a situation where we have only single bonds to this carbon , we 're talking about an sp3-hybridized carbon here , so we 're talking about a carbon-hydrogen bond where the carbon is sp3-hybridized . the signal for this carbon-hydrogen bond stretch shows up about 2900 wavenumbers . and so , how do we explain these different wavenumbers , because they 're all carbon-hydrogen bonds ? well , we need to think about the hybridizations . so let 's do that . so if we look at the sp-hybridized carbon , remember , that means that this carbon has two sp-hybrid orbitals . and an sp-hybrid orbital has the most s character out of all these orbitals we 've discussed here . so , actually 50 % s character , if you remember that from the videos on hybridizations . so 50 % s character for an sp-hybridized orbital . for an sp2-hybridized orbital , it 's about 33 % s character . and finally for an sp3-hybridized orbital , it 's about 25 % s character . and so going back to the sp-hybridized carbon , so the sp-hybrid orbital is 50 % s character . that means -- remember what that means . the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . and if this is the shortest bond , it must be the strongest bond out of these three that we 're talking about . so this carbon-hydrogen bond , where the carbon is sp-hybridized , is stronger that the carbon-hydrogen bond where the carbon is sp2-hybridized . this bond , though , has more s character than this one , so this bond is stronger than this one . so this order of bond strength explains the wavenumbers , because if you remember from the previous video , the bond strength affects the force constant , or the spring constant k , so as you increase in bond strength , you increase k , and we saw that increasing k increases the frequency , or the wavenumber . so this increases the frequency of bond vibrations , or increases the wavenumber where you would find the signal on your ir spectrum . and so since this is the strongest bond , this is the highest value for the wavenumber , so we 're going to find this signal more to the left on our ir spectrum when we 're looking at it . alright , now that we understand this idea , so the hybridization , we can look at some ir spectra for hydrocarbons , and we can analyze those . let 's do that . first , let 's compare alkanes and alkynes . let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . the second one is for 1-octyne , so a triple bond in this molecule . let 's compare these two , so you think about the differences . alright , one of the things that 's sometimes helpful to do is to draw a line around 3000 . so let me draw a line around 3000 . i 'm going to try and draw it for both here too . so i 'm going to draw a line around 3000 for both , so we can compare these two spectra . alright , we know that a carbon-hydrogen -- let me go and write this in here , a carbon-hydrogen bond where the carbon is sp3-hybridized , so that signal for that stretch shows up under 3000 . so that 's why it 's helpful to draw a line around 3000 . so that 's what we 're talking about when we 're talking about this complicated-looking thing in here . so it 's not really worth your time to analyze this in great detail , and of course my drawing of it is n't perfect to being with , but think about under 3000 , that 's where you expect to find your signal for your carbon-hydrogen bond , we 're talking about an sp3-hybridized carbon . and those are the only types of carbons that we have in decane . so , if we think about the diagnostic region versus the fingerprint region , so if i draw a line here to separate those two regions , in the diagnostic region , all we have is this . so all we 're thinking about is carbon-hydrogen where the carbon is sp3-hybridized . so very simple spectrum to analyze . we move on to 1-octyne , so now we 're looking at this one down here . so we see that same kind of thing , because obviously we have carbon-hydrogen sp3-hybridized also in this molecule . and so this is n't going to really help you too much when you 're analyzing the spectra , but it 's useful to know what you 're looking at , drawing a line at 3000 , and thinking about that 's what that represents . so once you draw a line at 3000 , it allows you to see some differences . so for example , this signal right here , if we drop down , it 's pretty close to 3300 , so this will be 3100 , 3200 , so 3300 . so approximately 3300 wavenumbers . and we know what that signal represents . we can go back up to here , we can look at about 3300 is where we would expect to see this carbon-hydrogen bond stretch where that carbon is sp-hybridized . so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . we also have a carbon right here , so that gives you your eight carbons . and so this bond , let me go and highlight it here , so this bond right here , this carbon-hydrogen bond , where this carbon is sp-hybridized , that 's this signal on our spectrum . so once again , it 's useful for analyzing here . alright , we have something else that shows up in the diagnostic region for this alkyne , and it 's this signal right here . so if we drop down , what 's the wavenumber where this signal appears ? the wavenumber is about 2100 , so approximately 2100 , maybe a little bit higher than that . and that 's the carbon triple bond stretch , so that 's the carbon-carbon triple bond stretch that we talked about in an earlier video . so that 's approximately the triple bond region when you 're looking at your spectrum . and of course , obviously we have one . this is an alkyne here . and so hopefully , this just shows you the differences , and once again , your fingerprint region over here is unique for each of these molecules here . so this shows you the differences and helps you to think about how to analyze your ir spectrum . let 's look at two more . actually , let 's look at one more here , and let 's compare these two . so now we have a spectrum for an alkene . so here we have 1-hexene , and let 's see if we can analyze this one . so we 're going to do the same thing , we 're going to draw a line around 1500 , so draw a line around 1500 , we 're going to draw a line around 3000 , and let 's analyze this one . so we know what this one is talking about , we know it 's talking about a carbon-hydrogen , where the carbon is sp3-hybridized . but what 's this signal right here ? we drop down , and that 's pretty close to 3100 . so that signal is approximately 3100 . and so , we know what that must represent . that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here . so this signal shows up around 3100 . so i draw a line up here , and we saw this signal at 3300 . so it 's useful to think about , it helps you distinguish on your ir spectra , if you draw a line in there , and think about where the wavenumber is . at what wavenumber does this signal appear ? we also have something else showing up in this one , let me go ahead and draw this one in here . what is this ? what is this guy right here ? that looks like a pretty obvious signal . so we can drop down here , and check out approximately where does that show up ? what 's that wavenumber ? well this is 1500 , this is 1600 , this is 1700 , so that 's a little bit -- that 's pretty much in between , it 's approximately 1650 . and in an earlier video , we said that was in the double bond region . so that 's the carbon-carbon double bond stretch signal , right in here . and obviously , there 's a double bond in our molecule . again , comparing these two , remember , we talked about the fact that a triple bond vibrates faster than a double bond . and so , the signal is different . the triple bond vibrates faster , so it has a higher wavenumber , the double bond does n't vibrate as fast , so it has a lower wavenumber . so these are important things to think about . finally , let 's compare this alkene to an arene here . so let 's look at toluene right here . alright , so if we do the same thing , if we draw a line around 3000 , so somewhere around 3000 . and we know this is below 3000 , so we know that this must be talking about carbon-hydrogen , where the carbon is sp3-hybridized . that 's a carbon-hydrogen bond stretch where the carbon is sp3-hybridized . well this carbon right here on toluene , so this is toluene , that carbon is sp3-hybridized , so that makes sense . we have this one little peak here , this one little signal that 's a little bit higher than 3000 , and so it 's pretty close to 3100 . and we know that 's approximately where we would find a carbon-hydrogen bond stretch where the carbon is sp2-hybridized . so somewhere around 3100 . and so , this makes this -- at first you might think , `` oh , well how do we tell this apart ? '' so this is very similar to this situation . so this looks very similar to this . and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . i 'm going to draw a line down here , and let 's try to connect the 1600s right here like that . and let 's think about what happened . so here 's the signal for the carbon-carbon double bond here , and when you 're talking about an aromatic double bond stretch , so carbon-carbon aromatic , that usually shows up lower than 1600 . so here it looks like we have two signals , so it depends on what kind of compound you 're dealing with . but we can see that this usually shows up lower . so this is somewhere usually around 1600 to 1450 . but we 're talking about the carbon-carbon double bond stretching here . and there 's some other subtle things that can clue you in , like this right here . so we do n't really see that on this one , on this spectrum . and then , these down here , we 're missing those here too . it would be way too much to get into in this video , to talk about what these mean , and that 's a little bit more than what we 've talked about so far , so that 'll have to be a different video . but there are subtle differences , and the easiest one to think about is to think about the fact that this aromatic carbon-carbon double bond shows up at a lower wavenumber than the one that we talked about right here . so just look at the sheer multitude of signals and sometimes that clues you into the fact that you 're dealing with this benzene ring here for toluene . so that sums up just a quick intro to looking at ir spectra for hydrocarbons .
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and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one .
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what is the double bond region range ?
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we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends on the hybridization state of this carbon . so let 's look at some examples here . so , if we look at an example where the carbon is sp-hybridized , so we know this is an sp-hybridized carbon because we have a triple bond here , so we 're talking about a carbon-hydrogen bond , where the carbon is sp-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3300 wavenumbers . if we look at this next example here , so now we have a carbon that has a double bond to it , so it must be sp2-hybridized , so we 're talking about a carbon-hydrogen bond , where the carbon is sp2-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3100 wavenumbers . and then finally , if we look at a situation where we have only single bonds to this carbon , we 're talking about an sp3-hybridized carbon here , so we 're talking about a carbon-hydrogen bond where the carbon is sp3-hybridized . the signal for this carbon-hydrogen bond stretch shows up about 2900 wavenumbers . and so , how do we explain these different wavenumbers , because they 're all carbon-hydrogen bonds ? well , we need to think about the hybridizations . so let 's do that . so if we look at the sp-hybridized carbon , remember , that means that this carbon has two sp-hybrid orbitals . and an sp-hybrid orbital has the most s character out of all these orbitals we 've discussed here . so , actually 50 % s character , if you remember that from the videos on hybridizations . so 50 % s character for an sp-hybridized orbital . for an sp2-hybridized orbital , it 's about 33 % s character . and finally for an sp3-hybridized orbital , it 's about 25 % s character . and so going back to the sp-hybridized carbon , so the sp-hybrid orbital is 50 % s character . that means -- remember what that means . the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . and if this is the shortest bond , it must be the strongest bond out of these three that we 're talking about . so this carbon-hydrogen bond , where the carbon is sp-hybridized , is stronger that the carbon-hydrogen bond where the carbon is sp2-hybridized . this bond , though , has more s character than this one , so this bond is stronger than this one . so this order of bond strength explains the wavenumbers , because if you remember from the previous video , the bond strength affects the force constant , or the spring constant k , so as you increase in bond strength , you increase k , and we saw that increasing k increases the frequency , or the wavenumber . so this increases the frequency of bond vibrations , or increases the wavenumber where you would find the signal on your ir spectrum . and so since this is the strongest bond , this is the highest value for the wavenumber , so we 're going to find this signal more to the left on our ir spectrum when we 're looking at it . alright , now that we understand this idea , so the hybridization , we can look at some ir spectra for hydrocarbons , and we can analyze those . let 's do that . first , let 's compare alkanes and alkynes . let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . the second one is for 1-octyne , so a triple bond in this molecule . let 's compare these two , so you think about the differences . alright , one of the things that 's sometimes helpful to do is to draw a line around 3000 . so let me draw a line around 3000 . i 'm going to try and draw it for both here too . so i 'm going to draw a line around 3000 for both , so we can compare these two spectra . alright , we know that a carbon-hydrogen -- let me go and write this in here , a carbon-hydrogen bond where the carbon is sp3-hybridized , so that signal for that stretch shows up under 3000 . so that 's why it 's helpful to draw a line around 3000 . so that 's what we 're talking about when we 're talking about this complicated-looking thing in here . so it 's not really worth your time to analyze this in great detail , and of course my drawing of it is n't perfect to being with , but think about under 3000 , that 's where you expect to find your signal for your carbon-hydrogen bond , we 're talking about an sp3-hybridized carbon . and those are the only types of carbons that we have in decane . so , if we think about the diagnostic region versus the fingerprint region , so if i draw a line here to separate those two regions , in the diagnostic region , all we have is this . so all we 're thinking about is carbon-hydrogen where the carbon is sp3-hybridized . so very simple spectrum to analyze . we move on to 1-octyne , so now we 're looking at this one down here . so we see that same kind of thing , because obviously we have carbon-hydrogen sp3-hybridized also in this molecule . and so this is n't going to really help you too much when you 're analyzing the spectra , but it 's useful to know what you 're looking at , drawing a line at 3000 , and thinking about that 's what that represents . so once you draw a line at 3000 , it allows you to see some differences . so for example , this signal right here , if we drop down , it 's pretty close to 3300 , so this will be 3100 , 3200 , so 3300 . so approximately 3300 wavenumbers . and we know what that signal represents . we can go back up to here , we can look at about 3300 is where we would expect to see this carbon-hydrogen bond stretch where that carbon is sp-hybridized . so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . we also have a carbon right here , so that gives you your eight carbons . and so this bond , let me go and highlight it here , so this bond right here , this carbon-hydrogen bond , where this carbon is sp-hybridized , that 's this signal on our spectrum . so once again , it 's useful for analyzing here . alright , we have something else that shows up in the diagnostic region for this alkyne , and it 's this signal right here . so if we drop down , what 's the wavenumber where this signal appears ? the wavenumber is about 2100 , so approximately 2100 , maybe a little bit higher than that . and that 's the carbon triple bond stretch , so that 's the carbon-carbon triple bond stretch that we talked about in an earlier video . so that 's approximately the triple bond region when you 're looking at your spectrum . and of course , obviously we have one . this is an alkyne here . and so hopefully , this just shows you the differences , and once again , your fingerprint region over here is unique for each of these molecules here . so this shows you the differences and helps you to think about how to analyze your ir spectrum . let 's look at two more . actually , let 's look at one more here , and let 's compare these two . so now we have a spectrum for an alkene . so here we have 1-hexene , and let 's see if we can analyze this one . so we 're going to do the same thing , we 're going to draw a line around 1500 , so draw a line around 1500 , we 're going to draw a line around 3000 , and let 's analyze this one . so we know what this one is talking about , we know it 's talking about a carbon-hydrogen , where the carbon is sp3-hybridized . but what 's this signal right here ? we drop down , and that 's pretty close to 3100 . so that signal is approximately 3100 . and so , we know what that must represent . that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here . so this signal shows up around 3100 . so i draw a line up here , and we saw this signal at 3300 . so it 's useful to think about , it helps you distinguish on your ir spectra , if you draw a line in there , and think about where the wavenumber is . at what wavenumber does this signal appear ? we also have something else showing up in this one , let me go ahead and draw this one in here . what is this ? what is this guy right here ? that looks like a pretty obvious signal . so we can drop down here , and check out approximately where does that show up ? what 's that wavenumber ? well this is 1500 , this is 1600 , this is 1700 , so that 's a little bit -- that 's pretty much in between , it 's approximately 1650 . and in an earlier video , we said that was in the double bond region . so that 's the carbon-carbon double bond stretch signal , right in here . and obviously , there 's a double bond in our molecule . again , comparing these two , remember , we talked about the fact that a triple bond vibrates faster than a double bond . and so , the signal is different . the triple bond vibrates faster , so it has a higher wavenumber , the double bond does n't vibrate as fast , so it has a lower wavenumber . so these are important things to think about . finally , let 's compare this alkene to an arene here . so let 's look at toluene right here . alright , so if we do the same thing , if we draw a line around 3000 , so somewhere around 3000 . and we know this is below 3000 , so we know that this must be talking about carbon-hydrogen , where the carbon is sp3-hybridized . that 's a carbon-hydrogen bond stretch where the carbon is sp3-hybridized . well this carbon right here on toluene , so this is toluene , that carbon is sp3-hybridized , so that makes sense . we have this one little peak here , this one little signal that 's a little bit higher than 3000 , and so it 's pretty close to 3100 . and we know that 's approximately where we would find a carbon-hydrogen bond stretch where the carbon is sp2-hybridized . so somewhere around 3100 . and so , this makes this -- at first you might think , `` oh , well how do we tell this apart ? '' so this is very similar to this situation . so this looks very similar to this . and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . i 'm going to draw a line down here , and let 's try to connect the 1600s right here like that . and let 's think about what happened . so here 's the signal for the carbon-carbon double bond here , and when you 're talking about an aromatic double bond stretch , so carbon-carbon aromatic , that usually shows up lower than 1600 . so here it looks like we have two signals , so it depends on what kind of compound you 're dealing with . but we can see that this usually shows up lower . so this is somewhere usually around 1600 to 1450 . but we 're talking about the carbon-carbon double bond stretching here . and there 's some other subtle things that can clue you in , like this right here . so we do n't really see that on this one , on this spectrum . and then , these down here , we 're missing those here too . it would be way too much to get into in this video , to talk about what these mean , and that 's a little bit more than what we 've talked about so far , so that 'll have to be a different video . but there are subtle differences , and the easiest one to think about is to think about the fact that this aromatic carbon-carbon double bond shows up at a lower wavenumber than the one that we talked about right here . so just look at the sheer multitude of signals and sometimes that clues you into the fact that you 're dealing with this benzene ring here for toluene . so that sums up just a quick intro to looking at ir spectra for hydrocarbons .
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the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have .
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is a sterically hindered oh bond that has reduced h-bonding going to always show up at ~3600 ?
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we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends on the hybridization state of this carbon . so let 's look at some examples here . so , if we look at an example where the carbon is sp-hybridized , so we know this is an sp-hybridized carbon because we have a triple bond here , so we 're talking about a carbon-hydrogen bond , where the carbon is sp-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3300 wavenumbers . if we look at this next example here , so now we have a carbon that has a double bond to it , so it must be sp2-hybridized , so we 're talking about a carbon-hydrogen bond , where the carbon is sp2-hybridized , the signal for this carbon-hydrogen bond stretch shows up about 3100 wavenumbers . and then finally , if we look at a situation where we have only single bonds to this carbon , we 're talking about an sp3-hybridized carbon here , so we 're talking about a carbon-hydrogen bond where the carbon is sp3-hybridized . the signal for this carbon-hydrogen bond stretch shows up about 2900 wavenumbers . and so , how do we explain these different wavenumbers , because they 're all carbon-hydrogen bonds ? well , we need to think about the hybridizations . so let 's do that . so if we look at the sp-hybridized carbon , remember , that means that this carbon has two sp-hybrid orbitals . and an sp-hybrid orbital has the most s character out of all these orbitals we 've discussed here . so , actually 50 % s character , if you remember that from the videos on hybridizations . so 50 % s character for an sp-hybridized orbital . for an sp2-hybridized orbital , it 's about 33 % s character . and finally for an sp3-hybridized orbital , it 's about 25 % s character . and so going back to the sp-hybridized carbon , so the sp-hybrid orbital is 50 % s character . that means -- remember what that means . the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . and if this is the shortest bond , it must be the strongest bond out of these three that we 're talking about . so this carbon-hydrogen bond , where the carbon is sp-hybridized , is stronger that the carbon-hydrogen bond where the carbon is sp2-hybridized . this bond , though , has more s character than this one , so this bond is stronger than this one . so this order of bond strength explains the wavenumbers , because if you remember from the previous video , the bond strength affects the force constant , or the spring constant k , so as you increase in bond strength , you increase k , and we saw that increasing k increases the frequency , or the wavenumber . so this increases the frequency of bond vibrations , or increases the wavenumber where you would find the signal on your ir spectrum . and so since this is the strongest bond , this is the highest value for the wavenumber , so we 're going to find this signal more to the left on our ir spectrum when we 're looking at it . alright , now that we understand this idea , so the hybridization , we can look at some ir spectra for hydrocarbons , and we can analyze those . let 's do that . first , let 's compare alkanes and alkynes . let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . the second one is for 1-octyne , so a triple bond in this molecule . let 's compare these two , so you think about the differences . alright , one of the things that 's sometimes helpful to do is to draw a line around 3000 . so let me draw a line around 3000 . i 'm going to try and draw it for both here too . so i 'm going to draw a line around 3000 for both , so we can compare these two spectra . alright , we know that a carbon-hydrogen -- let me go and write this in here , a carbon-hydrogen bond where the carbon is sp3-hybridized , so that signal for that stretch shows up under 3000 . so that 's why it 's helpful to draw a line around 3000 . so that 's what we 're talking about when we 're talking about this complicated-looking thing in here . so it 's not really worth your time to analyze this in great detail , and of course my drawing of it is n't perfect to being with , but think about under 3000 , that 's where you expect to find your signal for your carbon-hydrogen bond , we 're talking about an sp3-hybridized carbon . and those are the only types of carbons that we have in decane . so , if we think about the diagnostic region versus the fingerprint region , so if i draw a line here to separate those two regions , in the diagnostic region , all we have is this . so all we 're thinking about is carbon-hydrogen where the carbon is sp3-hybridized . so very simple spectrum to analyze . we move on to 1-octyne , so now we 're looking at this one down here . so we see that same kind of thing , because obviously we have carbon-hydrogen sp3-hybridized also in this molecule . and so this is n't going to really help you too much when you 're analyzing the spectra , but it 's useful to know what you 're looking at , drawing a line at 3000 , and thinking about that 's what that represents . so once you draw a line at 3000 , it allows you to see some differences . so for example , this signal right here , if we drop down , it 's pretty close to 3300 , so this will be 3100 , 3200 , so 3300 . so approximately 3300 wavenumbers . and we know what that signal represents . we can go back up to here , we can look at about 3300 is where we would expect to see this carbon-hydrogen bond stretch where that carbon is sp-hybridized . so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . we also have a carbon right here , so that gives you your eight carbons . and so this bond , let me go and highlight it here , so this bond right here , this carbon-hydrogen bond , where this carbon is sp-hybridized , that 's this signal on our spectrum . so once again , it 's useful for analyzing here . alright , we have something else that shows up in the diagnostic region for this alkyne , and it 's this signal right here . so if we drop down , what 's the wavenumber where this signal appears ? the wavenumber is about 2100 , so approximately 2100 , maybe a little bit higher than that . and that 's the carbon triple bond stretch , so that 's the carbon-carbon triple bond stretch that we talked about in an earlier video . so that 's approximately the triple bond region when you 're looking at your spectrum . and of course , obviously we have one . this is an alkyne here . and so hopefully , this just shows you the differences , and once again , your fingerprint region over here is unique for each of these molecules here . so this shows you the differences and helps you to think about how to analyze your ir spectrum . let 's look at two more . actually , let 's look at one more here , and let 's compare these two . so now we have a spectrum for an alkene . so here we have 1-hexene , and let 's see if we can analyze this one . so we 're going to do the same thing , we 're going to draw a line around 1500 , so draw a line around 1500 , we 're going to draw a line around 3000 , and let 's analyze this one . so we know what this one is talking about , we know it 's talking about a carbon-hydrogen , where the carbon is sp3-hybridized . but what 's this signal right here ? we drop down , and that 's pretty close to 3100 . so that signal is approximately 3100 . and so , we know what that must represent . that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and the one we just talked about . so let me go and highlight this here . so this signal shows up around 3100 . so i draw a line up here , and we saw this signal at 3300 . so it 's useful to think about , it helps you distinguish on your ir spectra , if you draw a line in there , and think about where the wavenumber is . at what wavenumber does this signal appear ? we also have something else showing up in this one , let me go ahead and draw this one in here . what is this ? what is this guy right here ? that looks like a pretty obvious signal . so we can drop down here , and check out approximately where does that show up ? what 's that wavenumber ? well this is 1500 , this is 1600 , this is 1700 , so that 's a little bit -- that 's pretty much in between , it 's approximately 1650 . and in an earlier video , we said that was in the double bond region . so that 's the carbon-carbon double bond stretch signal , right in here . and obviously , there 's a double bond in our molecule . again , comparing these two , remember , we talked about the fact that a triple bond vibrates faster than a double bond . and so , the signal is different . the triple bond vibrates faster , so it has a higher wavenumber , the double bond does n't vibrate as fast , so it has a lower wavenumber . so these are important things to think about . finally , let 's compare this alkene to an arene here . so let 's look at toluene right here . alright , so if we do the same thing , if we draw a line around 3000 , so somewhere around 3000 . and we know this is below 3000 , so we know that this must be talking about carbon-hydrogen , where the carbon is sp3-hybridized . that 's a carbon-hydrogen bond stretch where the carbon is sp3-hybridized . well this carbon right here on toluene , so this is toluene , that carbon is sp3-hybridized , so that makes sense . we have this one little peak here , this one little signal that 's a little bit higher than 3000 , and so it 's pretty close to 3100 . and we know that 's approximately where we would find a carbon-hydrogen bond stretch where the carbon is sp2-hybridized . so somewhere around 3100 . and so , this makes this -- at first you might think , `` oh , well how do we tell this apart ? '' so this is very similar to this situation . so this looks very similar to this . and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . i 'm going to draw a line down here , and let 's try to connect the 1600s right here like that . and let 's think about what happened . so here 's the signal for the carbon-carbon double bond here , and when you 're talking about an aromatic double bond stretch , so carbon-carbon aromatic , that usually shows up lower than 1600 . so here it looks like we have two signals , so it depends on what kind of compound you 're dealing with . but we can see that this usually shows up lower . so this is somewhere usually around 1600 to 1450 . but we 're talking about the carbon-carbon double bond stretching here . and there 's some other subtle things that can clue you in , like this right here . so we do n't really see that on this one , on this spectrum . and then , these down here , we 're missing those here too . it would be way too much to get into in this video , to talk about what these mean , and that 's a little bit more than what we 've talked about so far , so that 'll have to be a different video . but there are subtle differences , and the easiest one to think about is to think about the fact that this aromatic carbon-carbon double bond shows up at a lower wavenumber than the one that we talked about right here . so just look at the sheer multitude of signals and sometimes that clues you into the fact that you 're dealing with this benzene ring here for toluene . so that sums up just a quick intro to looking at ir spectra for hydrocarbons .
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so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized .
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if the structure was not drawn before and we had to identify what the structure were to be , how would we know ?
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let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the function g. it 's above the function g. we can write h of x , we could write h of x , as being equal to f of x minus g of x and what we wan na do is we wan na find the rate at which h changes with respect to x . we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that . it 's within our capabilities , but it 's important to realize when you 're taking the ap test that you have a calculator at your disposal . and a calculator can numerically integrate and can numerically evaluate derivatives . and so when they actually want us to find the area or evaluate an integral , where they give the endpoints , or evaluate a derivative at a point , well that 's a pretty good sign that you could probably use your calculator here . and what 's extra good about this is we have already essentially inputted h of x in the previous steps and really in part a. i defined this function here , and this function is essentially h of x. i took the absolute value of it so it 's always positive over either region , but i could delete the absolute value if we want . delete that absolute value , then i have to get rid of that parentheses at the end . let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here . g of x is right over here , and notice we are subtracting it . so the y1 , as i 've defined it in my calculator , and i just pressed this y equals button right over here , that is my h of x . now i can go back to the other screen , and i can evaluate its derivative when x is equal to 1.8 . i go to math . i scroll down . we have n derivative right here and so click enter there . and then what are we gon na take the derivative of ? well the function y sub 1 that i 've already defined in my calculator . i can go to variables , y variables , it 's already selected function so i 'll just press enter . and i 'll select the function y sub 1 that i 've already defined . so i 'm taking the derivative of y sub 1 . i 'm taking the derivative with respect to x . and i 'm going to evaluate that derivative , when x is equal to 1.8 . that simple . and then i click enter . and there you have it . it 's approximately -3.812 . so approximately -3.812 . and we 're done . one thing you might appreciate from this entire question , and even the question 1 , is that they really wanted to make sure that you understand the underlying conceptual ideas behind derivatives and integrals . and if you understand the conceptual ideas , of how do you use them to solve problems , and you have your calculator at disposal , then these are not too hairy , that these can be done fairly quickly .
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we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that .
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how do i know it is h ' ( 1.8 ) ?
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let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the function g. it 's above the function g. we can write h of x , we could write h of x , as being equal to f of x minus g of x and what we wan na do is we wan na find the rate at which h changes with respect to x . we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that . it 's within our capabilities , but it 's important to realize when you 're taking the ap test that you have a calculator at your disposal . and a calculator can numerically integrate and can numerically evaluate derivatives . and so when they actually want us to find the area or evaluate an integral , where they give the endpoints , or evaluate a derivative at a point , well that 's a pretty good sign that you could probably use your calculator here . and what 's extra good about this is we have already essentially inputted h of x in the previous steps and really in part a. i defined this function here , and this function is essentially h of x. i took the absolute value of it so it 's always positive over either region , but i could delete the absolute value if we want . delete that absolute value , then i have to get rid of that parentheses at the end . let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here . g of x is right over here , and notice we are subtracting it . so the y1 , as i 've defined it in my calculator , and i just pressed this y equals button right over here , that is my h of x . now i can go back to the other screen , and i can evaluate its derivative when x is equal to 1.8 . i go to math . i scroll down . we have n derivative right here and so click enter there . and then what are we gon na take the derivative of ? well the function y sub 1 that i 've already defined in my calculator . i can go to variables , y variables , it 's already selected function so i 'll just press enter . and i 'll select the function y sub 1 that i 've already defined . so i 'm taking the derivative of y sub 1 . i 'm taking the derivative with respect to x . and i 'm going to evaluate that derivative , when x is equal to 1.8 . that simple . and then i click enter . and there you have it . it 's approximately -3.812 . so approximately -3.812 . and we 're done . one thing you might appreciate from this entire question , and even the question 1 , is that they really wanted to make sure that you understand the underlying conceptual ideas behind derivatives and integrals . and if you understand the conceptual ideas , of how do you use them to solve problems , and you have your calculator at disposal , then these are not too hairy , that these can be done fairly quickly .
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let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here .
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it makes sense when sal says it , but on a real ap test , how do i interpret the written question correctly to assume they are asking for the derivative ( h ' ( x ) ) ?
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let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the function g. it 's above the function g. we can write h of x , we could write h of x , as being equal to f of x minus g of x and what we wan na do is we wan na find the rate at which h changes with respect to x . we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that . it 's within our capabilities , but it 's important to realize when you 're taking the ap test that you have a calculator at your disposal . and a calculator can numerically integrate and can numerically evaluate derivatives . and so when they actually want us to find the area or evaluate an integral , where they give the endpoints , or evaluate a derivative at a point , well that 's a pretty good sign that you could probably use your calculator here . and what 's extra good about this is we have already essentially inputted h of x in the previous steps and really in part a. i defined this function here , and this function is essentially h of x. i took the absolute value of it so it 's always positive over either region , but i could delete the absolute value if we want . delete that absolute value , then i have to get rid of that parentheses at the end . let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here . g of x is right over here , and notice we are subtracting it . so the y1 , as i 've defined it in my calculator , and i just pressed this y equals button right over here , that is my h of x . now i can go back to the other screen , and i can evaluate its derivative when x is equal to 1.8 . i go to math . i scroll down . we have n derivative right here and so click enter there . and then what are we gon na take the derivative of ? well the function y sub 1 that i 've already defined in my calculator . i can go to variables , y variables , it 's already selected function so i 'll just press enter . and i 'll select the function y sub 1 that i 've already defined . so i 'm taking the derivative of y sub 1 . i 'm taking the derivative with respect to x . and i 'm going to evaluate that derivative , when x is equal to 1.8 . that simple . and then i click enter . and there you have it . it 's approximately -3.812 . so approximately -3.812 . and we 're done . one thing you might appreciate from this entire question , and even the question 1 , is that they really wanted to make sure that you understand the underlying conceptual ideas behind derivatives and integrals . and if you understand the conceptual ideas , of how do you use them to solve problems , and you have your calculator at disposal , then these are not too hairy , that these can be done fairly quickly .
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we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that .
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how do i know it is h ' ( 1.8 ) ?
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let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the function g. it 's above the function g. we can write h of x , we could write h of x , as being equal to f of x minus g of x and what we wan na do is we wan na find the rate at which h changes with respect to x . we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that . it 's within our capabilities , but it 's important to realize when you 're taking the ap test that you have a calculator at your disposal . and a calculator can numerically integrate and can numerically evaluate derivatives . and so when they actually want us to find the area or evaluate an integral , where they give the endpoints , or evaluate a derivative at a point , well that 's a pretty good sign that you could probably use your calculator here . and what 's extra good about this is we have already essentially inputted h of x in the previous steps and really in part a. i defined this function here , and this function is essentially h of x. i took the absolute value of it so it 's always positive over either region , but i could delete the absolute value if we want . delete that absolute value , then i have to get rid of that parentheses at the end . let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here . g of x is right over here , and notice we are subtracting it . so the y1 , as i 've defined it in my calculator , and i just pressed this y equals button right over here , that is my h of x . now i can go back to the other screen , and i can evaluate its derivative when x is equal to 1.8 . i go to math . i scroll down . we have n derivative right here and so click enter there . and then what are we gon na take the derivative of ? well the function y sub 1 that i 've already defined in my calculator . i can go to variables , y variables , it 's already selected function so i 'll just press enter . and i 'll select the function y sub 1 that i 've already defined . so i 'm taking the derivative of y sub 1 . i 'm taking the derivative with respect to x . and i 'm going to evaluate that derivative , when x is equal to 1.8 . that simple . and then i click enter . and there you have it . it 's approximately -3.812 . so approximately -3.812 . and we 're done . one thing you might appreciate from this entire question , and even the question 1 , is that they really wanted to make sure that you understand the underlying conceptual ideas behind derivatives and integrals . and if you understand the conceptual ideas , of how do you use them to solve problems , and you have your calculator at disposal , then these are not too hairy , that these can be done fairly quickly .
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let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here .
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it makes sense when sal says it , but on a real ap test , how do i interpret the written question correctly to assume they are asking for the derivative ( h ' ( x ) ) ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ?
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what does it mean for the green sample to have two fragments of different sizes ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that .
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why would there be one fragment larger than another ?
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today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out of different kinds of substances , such as agarose and polyacrylamide , both of which i 'll discuss later . and the electrophoresis part of it means that you need to have an electrical field passing through the gel to get the bands to move . so to create an electrical field , we have to have a cathode and an anode . the cathode is on this side . remember from general chemistry that at the cathode , reduction takes place , so you 'll have a negative charge . but on the other side , where the anode is , you 'll have a positive charge , because this is where oxidation is taking place . and to power this , you 'll need some kind of battery to connect them . and these are both usually also connected to a big box that contains the gel electrophoresis apparatus . in order for charge to flow across , you need something that will conduct electricity , so you have a buffer that 's usually composed of different ions . so this is completely covering this entire gel . and remember that it 's always there , but i 'm going to erase it just because it 's going to get a little messy for what i need to show you next . so next , you 'll load a sample of dna . in order to load your sample , you 'll need to take a pipette and put it into one if these wells . so usually you 'll want to make sure that you have a dye mixed in with your sample so that you can see it as it 's running . so i 'll show that as a pink line here , so i 've filled that well , followed by yellow here , and green here . so that 's just the color of the dye . it can really be any color you want . this will just help you track the movement of the bands later on . once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this . maybe the pink will have split up into a few bands , indicating that there 's a few different sized fragments in there . with the yellow , you might only see one band . and with the green , you might actually have two bands , but they 're so close together still that it 's kind of hard to tell . so how can you really see what 's going on ? the solution is to let this run for a longer period of time . and eventually , what you 'll see is shown here at the bottom . you 'll see that whatever sized fragments were in your original samples have effectively split up by their size . and note that in pink , you 'll always want to load something known as a dna ladder . a dna ladder is like a standard . this is something that you buy from a company , and they tell you exactly what sizes their fragments so that you can match them up to your unknown samples . so this could be 400 base pairs , 200 base pairs , and 100 base pairs . and as you 'll note , the smallest fragments travel the furthest . this is because the smallest things are really easy to push with the electrical field . but when you have such a big molecule , or rather a big dna fragment , it can be hard to move . so you can see that the 400 base pair does n't move too fast or too far . and what 's this tell us about our unknown yellow and green samples ? this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next step of your experiment , if you wanted to insert it into a plasmid or a vector , you could cut this out of the gel and use it for that . now let 's talk about the two kinds of gels that are most commonly used . the first is agarose , and the second is sds-page . so agarose is a gel that 's usually used for separating big pieces of dna . so if you think about the pore size in the agarose , it has pretty big pores , so imagine it looking kind of like this . the gel is pretty big . there 's big holes here , so that you 'll be able to separate out the big pieces of dna that come through . however , if you 're trying to separate out little pieces , it wo n't be that obvious , because they 'll all just race through these giant holes . so remember that this is for big dna fragments . usually , this is for dna that 's bigger than 50 base pairs . sds-page , on the other hand , can be used for very small things . so imagine that being a much finer weaving with smaller pores . although this can be used for small pieces of dna , it can also be used for proteins . you might be wondering , what does sds-page even stand for . the sds part is sodium dodecyl sulfate . this is a chemical agent that denatures proteins , disrupting any non-covalent interactions they may have . this makes it so that the charge of the proteins is n't a factor when they 're separating out onto the gel , and they 're only being separated strictly by size . the page part is polyacrylamide gel electrophoresis , or we 'll just leave it at ge . so polyacrylamide is the substance that gel 's made out . so how can we remember the difference between these two types of gels ? remember that sds-page is for small dna or protein cells . s for small , and s for sds . and agarose is for bigger fragments of dna . so today we 've talked about how you would setup a gel electrophoresis , why it works , and how you would want to pick the substance but your gel 's made of . if you were really doing this in the lab , now that you have your fragments of known size of dna or protein , you could either sequence them or use them in other molecular techniques .
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once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is that -- it 'll look something like this .
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i 'm very confused , i understand that dna travels towards the positive end but should n't the positive end be the cathode ?
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