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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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when you get low numbers like 1,2,3 how do you find the square root of it ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ?
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why is the symbol facing right instead of left ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
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is there such thing as a quad root ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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when are you supposed to times the the square root number by the number in the radicle ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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can there be the square root of rational numbers and not just integers ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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do square roots only apply to perfect squares ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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so , ummm ... what would be the square root of 48 ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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is there a square root of any decimal number , or vice versa ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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the book and khan academy all square root easy ones for understanding but how do you square root 576 without a calculator ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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could a square root be a decimal , or a fraction ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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what would be the square root of 3,2 , and 1 ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 .
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wait so a squar root is just multiplying that number by its self that many times ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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is the square root of 25 -- -- -- 5 or -5 ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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in the real world , what are square roots good for ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ?
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where did this symbol originate ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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given a large number , say in the thousands , not equal to a perfect square , what 's an easy way to find the approximate square ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
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how do you find cubed root ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
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also how do you find the side length using squar or cubed root ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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are there square roots for negative numbers ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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how to find the square root of 729 ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol .
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is there an explanation about the difference between -3^2 and ( -3 ) ^2 ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
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what is another name of principle root ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ?
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what is the difference between a radicand and a radical ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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is the plus or minus sign the same as the sign in the practice where the plus and minus are touching ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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could there be a square root of a fraction ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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is there a way to find square roots for irrational numbers ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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how can i simplify a square root ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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is a square root based on a regular square ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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and 40 sec it was stated that the square root of 9 is = to 9 is it not 9*9= to 81 ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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what is square root hour ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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how do you find the square root or cube root of large numbers ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine .
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when you put parentesies on a negative such as ( -x ) ^2 it is postitive , but what about when you do n't ?
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if you 're watching a movie and someone is attempting to do fancy mathematics on a chalkboard , you 'll almost always see a symbol that looks like this . this radical symbol . and this is used to show the square root and we 'll see other types of roots as well , but your question is , well , what does this thing actually mean ? and now that we know a little bit about exponents , we 'll see that the square root symbol or the root symbol or the radical is not so hard to understand . so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ? well , that 's the same thing as three times three and that 's going to be equal to nine . but what if we went the other way around ? what if we started with the nine , and we said , well , what times itself is equal to nine ? we already know that answer is three , but how could we use a symbol that tells us that ? so , as you can imagine , that symbol is going to be the radical here . so , we could write the square root of nine , and when you look at this way , you say , okay , what squared is equal to nine ? and you would say , well , this is going to be equal to , this is going to be equal to , three . and i want you to really look at these two equations right over here , because this is the essence of the square root symbol . if you say the square root of nine , you 're saying what times itself is equal to nine ? and , well , that 's going to be three . and three squared is equal to nine , i can do that again . i can do that many times . i can write four , four squared , is equal to 16 . well , what 's the square root of 16 going to be ? well , it 's going to be equal to four . let me do it again . actually , let me start with the square root . what is the square root of 25 going to be ? well , this is the number that times itself is going to be equal to 25 or the number , where if i were to square it , i 'd get to 25 . well , what number is that , well , that 's going to be equal to five . why , because we know that five squared is equal to , five squared is equal to 25 . now , i know that there 's a nagging feeling that some of you might be having , because if i were to take negative three , and square it , and square it i would also get positive nine , and the same thing if i were to take negative four and i were to square the whole thing , i would also get positive 16 , or negative five , and if i square that i would also get positive 25 . so , why could n't this thing right over here , why ca n't this square root be positive three or negative three ? well , depending on who you talk to , that 's actually a reasonable thing to think about . but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root . if someone wants the negative square root of nine , they might say something like this . they might say the negative , let me scroll up a little bit , they might say something like the negative square root of nine . well , that 's going to be equal to negative three . and what 's interesting about this is , well , if you square both sides of this , of this equation , if you were to square both sides of this equation , what do you get ? well negative , anything negative squared becomes a positive . and then the square root of nine squared , well , that 's just going to be nine . and on the right-hand side , negative three squared , well , negative three times negative three is positive nine . so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually . let me write this a little bit more algebraically now . if we were to write , if we were to write the principal root of nine is equal to x . this is , there 's only one possible x here that satisfies it , because the standard convention , what most mathematicians have agreed to view this radical symbol as , is that this is a principal square root , this is the positive square root , so there 's only one x here . there 's only one x that would satisfy this , and that is x is equal to three . now , if i were to write x squared is equal to nine , now , this is slightly different . x equals three definitely satisfies this . this could be x equals three , but the other thing , the other x that satisfies this is x could also be equal to negative three , 'cause negative three squared is also equal to nine . so , these two things , these two statements , are almost equivalent , although when you 're looking at this one , there 's two x 's that satisfy this one , while there 's only one x that satisfies this one , because this is a positive square root . if people wanted to write something equivalent where you would have two x 's that could satisfy it , you might see something like this . plus or minus square root of nine is equal to x , and now x could take on positive three or negative three .
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principal , principal square root . square root . and another way to think about it , it 's the positive , this is going to be the positive square root .
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how can the square root of -9 be -3 ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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when in our lives will we use disributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way .
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so for example 34 ( 12+2^4 ) is ( 34 x 12 ) + and 2 to the fourth power is 16 so it is ( 34 x 12 ) + ( 34 x 16 ) right ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses .
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why do we need to know the distributive property if we know how to multiply a one digit number by a two digit number ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction .
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how do you write 3 to the second power ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression .
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when will we use algebra in our lives besides school ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what 's the difference between the communicative property and the distributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how do we use the distributive property in the real world ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ?
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would the same rules apply when a variable has taken place of the number ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what is the most important thing in distributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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would the distributive property still apply ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it .
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why do you have to 4 x 8 + 4 x 3 if the question only asks 4 x 8 + 3 ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how do you use the distributive property for decimals ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first .
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what would 34 ( x-1 ) equal ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what is the difference between associative and distributive properties ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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why is math so important ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first .
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when variables are in there ... or did i get smth wrong ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it .
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if the question was 2 ( 8 ) , will the property still apply ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 .
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can you make the 98 into a different number ( 100 ) and then subtract 2 ( 100 - 2 ) ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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how many properties are there ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how is the distributive property of multiplication over addition easier than the normal way ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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does distributive property go in order ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is .
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i mean when do we decide to put things in parentheses ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how do you apply the distributive property when working with variable fraction , with a whole number with variables as the distributive number ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now .
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why do we use the * symbol for multiplication ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what is associative , commutative , and distributive ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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when in our lives will we use distributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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why was the distributive property made ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ?
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which expression is equivalent to -3 ( -7+2x ) ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how do you show distributive property of -7x 32 ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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does the distributive property only apply when the addition expression is in parentheses ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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does the distributive law and the distributive property mean the same thing ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction .
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when using the distributive property could n't you just multiply 11 by 4 instead of 4 ( 8+3 ) or is 8+3=11 the whole point of the distributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works .
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for example , a video explaining anything along the lines of : 4 ( -x + 5 ) - 3 ( b + b ) + x ( 2 - 3 ) ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 .
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i was also wondering how do you remember the differences between commutative , associative and distributive law of addition ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ?
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what would 4 ( 6x+7 ) -9 ( 3x ) be ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses .
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witch one is more efficent the first one or the second on ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what does the distributive property mean ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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how do i put in a mutiplication symbol ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how long does it take to finish a distributive property over addition problems for you ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what are distributive property over addition ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what is 8 x 42 using distributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression .
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is it possible to use distributive law on addition calculations ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what 's the difference between a law and a property in math ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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does the distributive property always work ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 .
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why is it called the distributive law of multiplication over addition ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 .
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my problem is `` distribute '' ( 3x+4 ) 5 the x is really throwing me off.. help please ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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how many numbers are there in world history ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 .
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why is khan such a color freak ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 .
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what is the distributive law of multiplication over addition ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it .
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what is 1,512 x -8,965 ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression .
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may we use negative numbers ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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why is it multiplication over addition ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction .
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how can you solve 3 h + ( - 7 h ) ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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what does the fox say ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now .
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is distrubutive law of multiplication easier than the regular way of calculating ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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since multiplication and divison are two sides of the same coin , and addition and subtraction are two sides of the same coin , does that mean that , since multiplication has a distributive property over addition , that division has a distributive property over subtraction ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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does distributive property involve the order of operations ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how do use negatives and exponents in the distributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression .
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what is the difference between the 'distributive law ' and the 'bodmas ' rule ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is .
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what happens if there is a+b ( c+d ) ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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jenna made 6 bracelets using 32 bradshaw each kayla made 7 bracelets using 29 beads each who used more beads ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now .
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why is there no multiplication sign ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it .
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something i do n't get is if they do n't ask you to show you work , why does is matter which one u use , 4 x 8 + 4 x 3 or 8 plus 3 x 4 ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way .
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how do you rewrite a times ( b+12 ) and you have to put in 44+48 into the problem ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law .
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is there any other symbol that would represent addition in an expression or equation other than the plus symbol ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 .
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how do you remember the differences between commutative , associative and distributive law of addition ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no !
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i 'm having trouble with a problem on my homework ... 2x=2 ( x+2 ) is suppose to equal 4+x < -- ( x to the second power ) what am i missing ?
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