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let 's define ourselves some sets . so let 's say the set a is composed of the numbers 1 . 3 . 5 , 7 , and 18 . let 's say that the set b -- let me do this in a different color -- let 's say that the set b is composed of 1 , 7 , and 18 . and let 's say that the set c is composed of 18 , 7 , 1 , and 19 . now what i want to start thinking about in this video is the notion of a subset . so the first question is , is b a subset of a ? and there you might say , well , what does subset mean ? well , you 're a subset if every member of your set is also a member of the other set . so we actually can write that b is a subset -- and this is a notation right over here , this is a subset -- b is a subset of a . b is a subset . so let me write that down . b is subset of a . every element in b is a member of a . now we can go even further . we can say that b is a strict subset of a , because b is a subset of a , but it does not equal a , which means that there are things in a that are not in b . so we could even go further and we could say that b is a strict or sometimes said a proper subset of a . and the way you do that is , you could almost imagine that this is kind of a less than or equal sign , and then you kind of cross out this equal part of the less than or equal sign . so this means a strict subset , which means everything that is in b is a member a , but everything that 's in a is not a member of b . so let me write this . this is b . b is a strict or proper subset . so , for example , we can write that a is a subset of a . in fact , every set is a subset of itself , because every one of its members is a member of a . we can not write that a is a strict subset of a . this right over here is false . so let 's give ourselves a little bit more practice . can we write that b is a subset of c ? well , let 's see . c contains a 1 , it contains a 7 , it contains an 18 . so every member of b is indeed a member c. so this right over here is true . now , can we write that c is a subset ? can we write that c is a subset of a ? can we write c is a subset of a ? let 's see . every element of c needs to be in a . so a has an 18 , it has a 7 , it has a 1 . but it does not have a 19 . so once again , this right over here is false . now we could have also added -- we could write b is a subset of c. or we could even write that b is a strict subset of c. now , we could also reverse the way we write this . and then we 're really just talking about supersets . so we could reverse this notation , and we could say that a is a superset of b , and this is just another way of saying that b is a subset of a . but the way you could think about this is , a contains every element that is in b . and it might contain more . it might contain exactly every element . so you can kind of view this as you kind of have the equals symbol there . if you were to view this as greater than or equal . they 're note quite exactly the same thing . but we know already that we could also write that a is a strict superset of b , which means that a contains everything b has and then some . a is not equivalent to b . so hopefully this familiarizes you with the notions of subsets and supersets and strict subsets .
so we actually can write that b is a subset -- and this is a notation right over here , this is a subset -- b is a subset of a . b is a subset . so let me write that down .
does it hold true that if a and b are sets , and b is a strict subset of a , then b can not be equal nor equivalent to a ?
let 's define ourselves some sets . so let 's say the set a is composed of the numbers 1 . 3 . 5 , 7 , and 18 . let 's say that the set b -- let me do this in a different color -- let 's say that the set b is composed of 1 , 7 , and 18 . and let 's say that the set c is composed of 18 , 7 , 1 , and 19 . now what i want to start thinking about in this video is the notion of a subset . so the first question is , is b a subset of a ? and there you might say , well , what does subset mean ? well , you 're a subset if every member of your set is also a member of the other set . so we actually can write that b is a subset -- and this is a notation right over here , this is a subset -- b is a subset of a . b is a subset . so let me write that down . b is subset of a . every element in b is a member of a . now we can go even further . we can say that b is a strict subset of a , because b is a subset of a , but it does not equal a , which means that there are things in a that are not in b . so we could even go further and we could say that b is a strict or sometimes said a proper subset of a . and the way you do that is , you could almost imagine that this is kind of a less than or equal sign , and then you kind of cross out this equal part of the less than or equal sign . so this means a strict subset , which means everything that is in b is a member a , but everything that 's in a is not a member of b . so let me write this . this is b . b is a strict or proper subset . so , for example , we can write that a is a subset of a . in fact , every set is a subset of itself , because every one of its members is a member of a . we can not write that a is a strict subset of a . this right over here is false . so let 's give ourselves a little bit more practice . can we write that b is a subset of c ? well , let 's see . c contains a 1 , it contains a 7 , it contains an 18 . so every member of b is indeed a member c. so this right over here is true . now , can we write that c is a subset ? can we write that c is a subset of a ? can we write c is a subset of a ? let 's see . every element of c needs to be in a . so a has an 18 , it has a 7 , it has a 1 . but it does not have a 19 . so once again , this right over here is false . now we could have also added -- we could write b is a subset of c. or we could even write that b is a strict subset of c. now , we could also reverse the way we write this . and then we 're really just talking about supersets . so we could reverse this notation , and we could say that a is a superset of b , and this is just another way of saying that b is a subset of a . but the way you could think about this is , a contains every element that is in b . and it might contain more . it might contain exactly every element . so you can kind of view this as you kind of have the equals symbol there . if you were to view this as greater than or equal . they 're note quite exactly the same thing . but we know already that we could also write that a is a strict superset of b , which means that a contains everything b has and then some . a is not equivalent to b . so hopefully this familiarizes you with the notions of subsets and supersets and strict subsets .
and there you might say , well , what does subset mean ? well , you 're a subset if every member of your set is also a member of the other set . so we actually can write that b is a subset -- and this is a notation right over here , this is a subset -- b is a subset of a .
is n't every thing a strict subset of the universal set ?
let 's define ourselves some sets . so let 's say the set a is composed of the numbers 1 . 3 . 5 , 7 , and 18 . let 's say that the set b -- let me do this in a different color -- let 's say that the set b is composed of 1 , 7 , and 18 . and let 's say that the set c is composed of 18 , 7 , 1 , and 19 . now what i want to start thinking about in this video is the notion of a subset . so the first question is , is b a subset of a ? and there you might say , well , what does subset mean ? well , you 're a subset if every member of your set is also a member of the other set . so we actually can write that b is a subset -- and this is a notation right over here , this is a subset -- b is a subset of a . b is a subset . so let me write that down . b is subset of a . every element in b is a member of a . now we can go even further . we can say that b is a strict subset of a , because b is a subset of a , but it does not equal a , which means that there are things in a that are not in b . so we could even go further and we could say that b is a strict or sometimes said a proper subset of a . and the way you do that is , you could almost imagine that this is kind of a less than or equal sign , and then you kind of cross out this equal part of the less than or equal sign . so this means a strict subset , which means everything that is in b is a member a , but everything that 's in a is not a member of b . so let me write this . this is b . b is a strict or proper subset . so , for example , we can write that a is a subset of a . in fact , every set is a subset of itself , because every one of its members is a member of a . we can not write that a is a strict subset of a . this right over here is false . so let 's give ourselves a little bit more practice . can we write that b is a subset of c ? well , let 's see . c contains a 1 , it contains a 7 , it contains an 18 . so every member of b is indeed a member c. so this right over here is true . now , can we write that c is a subset ? can we write that c is a subset of a ? can we write c is a subset of a ? let 's see . every element of c needs to be in a . so a has an 18 , it has a 7 , it has a 1 . but it does not have a 19 . so once again , this right over here is false . now we could have also added -- we could write b is a subset of c. or we could even write that b is a strict subset of c. now , we could also reverse the way we write this . and then we 're really just talking about supersets . so we could reverse this notation , and we could say that a is a superset of b , and this is just another way of saying that b is a subset of a . but the way you could think about this is , a contains every element that is in b . and it might contain more . it might contain exactly every element . so you can kind of view this as you kind of have the equals symbol there . if you were to view this as greater than or equal . they 're note quite exactly the same thing . but we know already that we could also write that a is a strict superset of b , which means that a contains everything b has and then some . a is not equivalent to b . so hopefully this familiarizes you with the notions of subsets and supersets and strict subsets .
so , for example , we can write that a is a subset of a . in fact , every set is a subset of itself , because every one of its members is a member of a . we can not write that a is a strict subset of a .
is one significantly more common than the other ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name .
how long after birth does the foramen ovale remain operative ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel .
how does the connection between pulmonary arota and vein disappear after childbirth ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs .
since the artery does n't have any valves , would the blood leak from the aorta to the left atrium through the ductus arteriosus ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
why is the heart purple ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale .
ca n't blood just flow through the lungs regularly without having to find a shortcut ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel .
would it be correct in assuming that the remaining 10 % of blood that travels into the lungs provide the lungs with the appropriate oxygen and nutrients needed for the lungs themselves during fetal development ( and by extension , at least nutrient transport in adult lungs ) , or is this from an entirely different pulmonary blood supply system ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place .
since the aorta is positioned more on the left side of your heart , does blood only from the left pulmonary artery get to use the ductus arteriosus ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar .
so how does the fetus gets oxygenated blood ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn .
is the entire pathway between the right atrium and the left atrium called the foramen ovale , or does that term only refer to the hole in the septum secundum ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
why is it important for the fetal heart to have both the ductus arterioss and foramen ovale , but not good for the adult heart to have these ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung .
so thats why lungs are developed last ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over .
i am thinking of persistent fetal circulation with vsd ... .. whats your opinion ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over .
does it always remain open ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name .
is that hole the foramen ovale which probably did n't close ?
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over . the first thing i want you to notice is that it 's mostly got what i 've drawn is kind of this purplish blood . in the adult heart we know there 's a real clear distinction between oxygenated blood and deoxygenated blood . but in the fetal heart it 's are all very , very similar . now let 's start by kind of orienting ourselves . this vessel at the top is different , right ? it 's got blue blood rather than this kind of purplish blood . and the reason for that is that this blood is actually coming back from the body , and the body has used up as much oxygen as it can . so this is the superior vena cava dragging blood back from the arms and specifically the head region . and you 've also got on the bottom , blood coming into the heart from the inferior vena cava . now , this is also coming from the body , but i 've drawn it more pink . so why would i do that ? well , it 's because you remember there 's also , in addition to just bringing blood from the body , there 's also blood coming from the umbilical vein . and i do n't want you to forget that , because the umbilical vein is actually bringing really , really oxygen-rich blood from the placenta and it 's mixing with the inferior vena cava . so it 's not bright red , but it 's got this kind of pinkish tone to it . and this is really the only major source of oxygen that the fetus is getting is from this umbilical vein . so this is actually very , very important . and that 's why when it mixes with that blue blood from the superior vena cava in the right atrium , we get kind of this purplish stuff . and of course -- let me just quickly label the rest of the chambers . you got the right ventricle , the left atrium , and the left ventricle . and these are the four chambers . let me also name the major arteries and veins . this , of course , is the aorta at the top . i 've also got the pulmonary artery here -- pulmonary artery . and we 've got pulmonary veins . and i 'm just going to label this side right there pulmonary veins . but you see there are two on the other side as well . so this is what the fetal heart looks like , and now let 's actually think about what 's going on in the fetal heart and how the blood is flowing through . so to do that , let me actually start out by drawing some lungs , because this is actually going to help inform the path of blood . so these are the lungs . let 's say this is the right lung . and of course , there is one on the left as well . let me just draw it in just so we do n't forget that it exists . but i 'm going to use the right lung for this example . this is our left lung . i even drew the little cardiac notch . and on the right lung , blood is coming in from , let 's say , the pulmonary artery , right ? so blood is coming in this way from the pulmonary artery . it 's going to go into little vessels , little arterioles . i 'll draw them kind of the same purplish color . it 's going to go into the little arterioles . and then it 's going to get into little capillaries . even tinier little blood vessels . and those capillaries are going to go and meet up with an alveolar sac . and these sacs in adults are full of air . but in the fetus there 's actually nothing but fluid inside of here . so it 's actually just full of amniotic fluid . so it 's just fluid filled . and so if you 're thinking about it , do you expect the oxygen level to be high or low ? well , if it 's full of fluid , amniotic fluid , it 's going to be pretty low . there 's not much oxygen there . in fact , blood is n't even going to the lungs to get oxygen , because we said that the main source of oxygen for the fetus is going to be from the umbilical vein . this is where the vast majority of oxygen is coming from . so what ends up happening is that because there 's such low oxygen in the alveolar sacs , they have this process , this ability , to actually cause the arterioles to constrict . these arterioles have some smooth muscle on them . and the alveolar sacs , because of the low oxygen , they make this constrict . so it literally kind of clamps down and it looks a little tighter -- something like this , a skinnier blood vessel . and when it constricts , when you have a smaller radius on that blood vessel , what does that mean exactly ? well , it means that the amount of resistance went up here . and of course , if it happens once , that 's not a big deal . but if it happens in millions and millions of arterioles all throughout the lungs , then what we 're really talking about is that the pulmonary artery -- both of them on both sides -- are going to face really high resistance . and this process of kind of increasing the resistance when the amount of oxygen is low -- remember , we actually named this process . this process is called hypoxic , and that just means low oxygen . pulmonary , referring to the lungs -- hypoxic pulmonary . vaso , meaning blood vessel . constriction , so making the blood vessels tight . so this is the process that we 're talking about , hypoxic pulmonary vasoconstriction . and it happens in adults , but it also happens in the fetus . and it is actually very important in the fetus , because the lungs , both the left and right , are full of fluid it allows the blood vessels to constrict . i say allows , but it causes , let 's say , the blood vessels to constrict . and it really raises the amount of resistance that the pulmonary arteries are facing . now , if they 're facing a lot of resistance , think about what that means . that means that if the heart wants to pump blood to the lungs , it 's going to have to raise the pressure . so the pressure goes up in the pulmonary artery . and if the pressure is high there , that means the pressure is going to be high in the right ventricle . and if the pressure is high in the right ventricle , blood has to get in there somehow so the pressures start going up in the right atrium . so pressures start going up everywhere . and so really what the heart faces is a choice . it can either continue to just try to push blood into the lungs , even though there 's a lot of resistance , or it can try to find a shortcut really to bypass the lungs all together . and that idea of shortcuts is really what we 're talking about in the fetal heart . in fact , there are two shortcuts to get blood from this side , the right side , either the right atrium or the right ventricle or the pulmonary artery , over to the left side . and when i say left side , i really mean at the end of the day the aorta , or i guess you could think of the left atrium or left ventricle as well . but really at the end of the day you want to get blood into the aorta . and you want to think of a clever way of doing it and being able to bypass the lungs . that 's the challenge . so how does that fetal heart meet that challenge ? how does it bypass the lungs ? two major ways . so let me draw them both out . i 'm going to start with drawing kind of a blow up of this section right here . let 's say i blow that up , and i 'm going to try to sketch it out here for you . let 's see if i can make it neat . this is kind of the same box . so i just want to make sure we 're not confused by the way i 'm drawing it . this is just a blow up of that section . and if you looked closely , what you would see is that there 's a wall , right ? the same wall that i drew . but there 's actually not just one wall , it 's two walls stuck together . that 's actually kind of the first point i want to make , is that there 's not just one , but two walls there . and this one is called septum primum . septum just kind of refers to a wall . and primum is the latin word for first . so septum primum is this wall over here on this side , this guy . and septum secundum is the other wall . so you 've got two walls next to each other . and they look almost like one wall , but there 's actually two . and that 's the septum secundum . and just to make sure we 're still kind of oriented to the right and left atrium , on this side is the right atrium and on this side is the left atrium . and remember , the kind of overall goal is to somehow bypass the lungs . and what i mean is , get blood from here somehow over to the other side . and what happens is that when you look closely at the septum secundum -- if you look closely at that one -- there 's a little tiny hole there . so imagine a piece of swiss cheese , and if i was looking at this wall as if it 's a piece of swiss cheese , the hole actually is in the wall . so there 's a little hole there . and so if i kind of stuck my finger in here -- let 's say i stuck my finger right here -- i would actually be able to touch -- from the right atrium i could actually touch the septum primum , because of the fact that there 's a hole in the wall . and so this hole -- let me take my finger out of here now -- this hole is called the foramen ovale . let me actually just label that for you , or foramen ovale . so the foramen ovale is this hole . so instead of calling it hole , you can now call it by its full name . and this foramen ovale is right there . now , it turns out the septum primum also is kind of like a piece of swiss cheese . both of them are like little pieces of swiss cheese . and it 's got a little break in its wall as well . so now think about it . what will happen if the pressure is really high in the right atrium ? pressures are really high on this side . and of course pressure is kind of force pushing out on the area , the surface area of the chamber . so blood is kind of pushing in all directions . and when it pushes here , right in the foramen ovale , what 's going to happen ? well , right at that spot , if there 's pressure , then this septum primum does an interesting thing . it becomes a little bit like a flap or a valve , and it kind of falls away . it falls away like that . and so -- bingo -- you 've got access , right ? right atrium blood is going to flow right into the left atrium , because the septum primum became a little flap and it fell away . so let me actually now re-sketch this out in the middle drawing , and make it the way it should be drawn . so instead of drawing it like that , you 've got literally a little flap here . this is the septum primum flap of tissue . and here i 'm drawing the septum secundum , right ? and i 'm going to draw blood now going through . so you 've got blood flowing through -- and let me make sure i got this right -- and right there . so blood is now going to go through from here into the left atrium . and remember this little hole -- i ca n't draw a hole very easily -- but you can imagine that there 's a hole there . in the wall of the septum secundum where the blood is going through that hole is called the foramen ovale . and so when i said there are a couple of tricks , this is trick number one . trick number one is getting blood from the right atrium directly over to the left atrium . because you know that once it gets over there , now it just literally has bypassed the lungs . but that 's only one of two tricks . so not all of the blood in the right atrium goes through the foramen ovale . not all of it . some blood actually passes through the normal way . it goes through the tricuspid valve into the right ventricle . and if blood is going to the right ventricle , that is a good thing . why ? because we want to make sure our right ventricle is pumping . we want to make sure that it 's squeezing , getting practice , and that those muscles are getting stronger . now , the right ventricle is going to do its job . it 's going to pump blood into the pulmonary arteries , the left and right pulmonary arteries . and that blood is facing , as i said before , a lot of resistance . so there 's another little trick . it turns out that there 's another place for the blood to go . you 're looking at this picture and you 're thinking , well , i do n't see any other place for blood to go . there 's only the left pulmonary artery and the right pulmonary artery . but there is another place . it turns out that the fetal heart actually has a little vessel here . i 'm going to do the vessel in another color . so it has a little vessel here . and this vessel allows blood to go through . so you can actually get blood to pass through this vessel like so . so this is actually really cool , right ? because you can now see how blood can go directly from the pulmonary artery into the aorta and go down . this is our trick number two . and so this little vessel , this little guy right here , i 'm going to loop it and name it . this is our ductus arteriosus . now , remember there 's another one called the ductus venosus . so this is ductus arteriosus , so a different name . but this is trick number two . so one trick was to go from the right atrium to the left atrium , and another trick was to go from the pulmonary arteries to the aorta . so these are the two major tricks . and you basically can see now that blood is going to bypass the lungs using either trick . now , does that mean that no blood goes to the lungs ? no . a little bit of blood does go to lungs . in fact , about 10 % or so continues to the lungs , but 90 % actually goes through one of these two pathways , either through the ductus or through the foramen ovale . so these are the kind of interesting differences between the fetal heart and the adult heart .
what you 're looking at is the fetal heart . it looks a lot like the adult heart , but a couple of interesting differences that we 're going to go over .
would n't blowing up part of the fetal heart result in killing the fetus ?
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 .
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 .
how can you get the square root of 4 ?
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 .
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 .
sal says that the principal square root is positive , but could it also be thought of as the absolute value of the square root ?
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 .
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 .
how can the square root of -9 be -3 ?
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 .
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 .
does this mean that exponent to the 2nd power and square root are the same thing ?
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 .
so , let 's start with an example . so , we know that three to the second power is what ? three squared is what ?
the same with exponents to the 3rd power and cube roots ?
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 .
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 ?
or is using a radical symbol not in simplest notation ?
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 .
but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
so is sqare root basically the same thing as exponents ?
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 .
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 .
what is i want to get negative squares and how do i square big numbers ?
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 .
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 .
what is the name of the square root sign ?
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 .
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 .
why ca n't the number being `` square rooted '' be negative ?
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 .
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 .
how do you solve a square root equation ?
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 .
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 .
for example 100 square root would be 10^2 but how would you figure it out if you did n't know that 10^2 equals 100 ?
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 .
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 .
what is the the square root of -9 ?
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 .
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 .
is there a difference between principle and perfect square roots ?
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 .
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 .
how do you calculate square roots without a calculator ?
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 .
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 .
if ( x+y ) ^2= 17 and xy=3 then the expression x^2+y^2 is equal to ?
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 .
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 .
what is the difference between a regular number squared and a number in brackets squared ?
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 .
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 ?
anyone have a solid universal way for solving perfect & non-perfect squares ?
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 .
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 .
when do you find a square root according to order of operations ?
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 .
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 .
what are some ways to figure out the square root ?
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 .
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 .
when u are trying to find the square root of a decimal , do u divide it even though the decimal is already squared ?
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 .
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 .
how do you find the square root of a negative ?
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 .
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 ?
what is a 'radical ' symbol ?
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 .
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 .
how can i do a square root and cube root without calculator ?
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 .
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 .
is there an easier way to find square roots like the square root of 1024 ?
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 .
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 ?
the second part of my earlier question is : if the two similar-looking symbols ( the radical or the long division symbol ) were meant to look similar , then does anyone happen to know which symbol came first ?
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 .
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 .
so ... are exponents and square roots the same thing ?
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 .
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 .
is square root the inverse function of f ( x ) =x^2 ?
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 .
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 .
why does this convention exist that when we take square root of a number we always imply `` principal square root `` ?
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 .
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 .
what is square root of -100 ?
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 .
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 .
if the square of all numbers is positive , what 's sqrt ( -1 ) ?
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 .
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 you have an odd number , such as 17 , how would you find a square root for it ?
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 .
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 .
how can we call the square root of 2 irrational when it can be expressed ?
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 .
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 .
is it true that the square root of 169 is both 13 and -13 ?
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 .
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 ?
is the slag term `` radical '' related at all ?
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 .
so , it all works out . nine is equal , nine is equal to nine . and so this is an interesting thing , actually .
how is negative nine squared equal to nine ?
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 .
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 .
how do you find the square root without a calculator ?
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 .
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 .
what is the square root of 169 how do you find that ?
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 .
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 .
then is 3 to the 3rd power the same as the square root of 27 ?
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 .
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 .
why is called square root ?
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 .
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 .
are spuare roots always perfect ?
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 .
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 .
how do you write the square root symbol on a keyboard ?
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 .
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 .
how can we , lets say solve , principal square root of 72 ?
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 .
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 .
so does that mean that any negative to the x power ( x being any number ) is equal to a positive ?
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 .
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 .
how do i find the square root of a number like 12356 that is large ?
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 .
but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
can there be a rational number that can be a root ?
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 .
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 .
can there be a negative decimal square root ?
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 .
but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
so there is a negative root ?
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 .
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 .
is the square root of negative nine the same thing as the negative square root of nine ?
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 .
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 .
i have a question ... if you have a number like 18 , would the square root be 6 or 3 ?
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 .
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 .
why would a negative number square rooted become a positive number ?
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 .
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 .
is there an easy way to find the square roots of larger numbers that you ca n't do with the help of mental math ?
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 .
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 .
so if we have the square root symbol and 24 , the answer could be 4 and -4 , right ?
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 .
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 .
when you are thinking of square and cube roots , do you think in dimensions ?
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 .
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 .
say squares are one dimension , cubes twos are three dimensions , 3s ?
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 .
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 .
is there a way to solve big square roots that are perfect squares that is somewhat like long division ?
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 .
but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
who invented the root sign ?
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 .
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 .
what is the best way to memorize square roots 1-30 ?
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 .
but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
is there a 1/2 root ?
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 .
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 .
what are some real-life examples of why you would need to find the square root of something ?
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 .
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 .
can a square root be negative ?
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 .
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 .
how do you find the square root of an odd number ?
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 .
but when you see a radical symbol like this , people usually call this the principal root . principal root . principal , principal square root .
what is a cube root ?
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 .
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 .
can you do square roots with negative numbers or not ?
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 .
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 .
can you round a square root for example 8.5 round it to 9 and find the square root ?
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 .
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 .
if negative 3 squared is positive 9 , how do you find the square root of negative 9 ?
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 .
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 .
how high can you go with roots ?
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 .
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 .
what is the least whole number greater than 1 that is both a perfect square and a perfect cube ?
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 .
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 .
how do you figure out the square root of 289 ?
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 .
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 .
can there be a square root of infinity ?
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 .
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 .
how can you solve square roots ?
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 .
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 .
for example , how can you solve the square root of a very large number , not something like nine ?
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 .
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 .
is there a specific formula to finding the square root of a number ?
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 .
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 you have an odd number , such as 17 , how would you find a square root for it ?
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 .
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 .
where did the square root symbol originate from ?
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 .
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 .
in the problem with the square root of 9 why was n't 9 multiplied by 2 ?
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 .
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 .
what is the square root of 64 ?
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 .
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 .
5 , what is a cube root and what is a square root ?
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 .
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 .
can someone re explain the concept of square roots and the negative square roots ?
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 .
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 .
what is a square root ?
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 .
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 .
what if you want to find the square root of a big number , for example , 196 ?
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 .
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 .
why does the square root help you ?