Video Transcripts: Square Roots

Intro to 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 couldn't  this thing right over here, why can'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. 

Understanding square roots 

We're asked to find the square root of 100. Let me write this down bigger. So the square root is  this big check-looking thing. The square root of 100. When you see it like this, this means the  positive square root. If you're familiar with negative numbers, you know that there's also a  negative square root, but when you just see this symbol, that means the positive square root.  So let's think about what this is saying. This is asking us find the number, the positive number,  that when I multiply that number by itself, I get 100. So what number when I multiply it by itself do I get 100? Well, let's see, if I multiply 9 by itself, that's only going to be 81. If I multiply 10 by  itself, that is 100. So this is equal to-- and let me write it this way. Normally, you could skip this  step. But you could write this as the square root of-- and instead of 100, 100 is the same thing  as 10 times 10. And then you know, the square root of something times itself, that's just going  to be that something. This is just equal to 10. So the square root of 100 is 10. Or another way  you could write, I guess, this same truth is that 10 squared, which is equal to 10 times 10, is  equal to 100. 

Square root of decimal 

Let's see if we can solve the equation P squared is equal to 0.81. So how could we think about  this? Well one thing we could do is we could say, look if P squared is equal to 0.81, another way  of expressing this is, that well, that means that P is going to be equal to the positive or negative square root of 0.81. Remember if we just wrote the square root symbol here, that means the  principal root, or just the positive square root. But here P could be positive or negative, because if you square it, if you square even a negative number, you're still going to get a positive value.  So we could write that P is equal to the plus or minus square root of 0.81, which kind of helps  us, it's another way of expressing the same, the same, equation. But still, what could P be? In  your brain, you might immediately say, well okay, you know if this was P squared is equal to 81,  I kinda know what's going on. Because I know that nine times nine is equal to 81. Or we could  write that nine squared is equal to 81, or we could write that nine is equal to the principal root  of 81. These are all, I guess, saying the same truth about the universe, but what about 0.81?  Well 0.81 has two digits behind, to the right of the decimal and so if I were to multiply  something that has one digit to the right of the decimal times itself, I'm gonna have something  with two digits to the right of the decimal. And so what happens if I take, instead of nine  squared, what happens if I take 0.9 squared? Let me try that out. Zero, I'm gonna use a  different color. So let's say I took 0.9 squared. 0.9 squared, well that's going to be 0.9 times 0.9, which is going to be equal to? Well nine times nine is 81, and I have one, two, numbers to the 

right of the decimal, so I'm gonna have two numbers to the right of the decimal in the product.  So one, two. So that indeed is equal to 0.81. In fact we could write 0.81 as 0.9 squared. So we  could write this, we could write that P is equal to the plus or minus, the square root of, instead  of writing 0.81, I could write that as 0.9 squared. In fact I could also write that as negative 0.9  squared. Cause if you put a negative here and a negative here, it's still not going to change the  value. A negative times a negative is going to be a positive. I could, actually I would have put a  negative there, which would have implied a negative here and a negative there. So either of  those are going to be true. But it's going to work out for us because we are taking the positive  and negative square root. So this is going to be, P is going to be equal to plus or minus 0.9. Plus  or minus 0.9, or we could write it that P is equal to 0.9, or P could be equal to negative 0.9. And  you can verify that, you would square either of these things, you get 0.81. 

Intro to cube roots 

We already know a little bit about square roots. For example, if I were to tell you that seven  squared is equal to 49, that's equivalent to saying that seven is equal to the square root of 49.  The square root essentially unwinds taking the square of something. In fact, we could write it  like this. We could write the square root of 49, so this is whatever number times itself is equal to 49. If I multiply that number times itself, if I square it, well I'm going to get 49. And that's going  to be true for any number, not just 49. If I write the square root of X and if I were to square it,  that's going to be equal to X and that's going to be true for any X for which we can evaluate the  square root, evaluate the principle root. Now typically and as you advance in math you're going to see that this will change, but typically you say, okay if I'm going to take the square root of  something, X has to be non-negative. X has to be non-negative. This is going to change once  we start thinking about imaginary and complex numbers, but typically for the principle square  root, we assume that whatever's under the radical, whatever's under here, is going to be non negative because it's hard to square a number at least the numbers that we know about, it's  hard to square them and get a negative number. So for this thing to be defined, for it to make  sense, it's typical to say that, okay we need to put a non-negative number in here. But anyway,  the focus of this video is not on the square root, it's really just to review things so we can start  thinking about the cube root. And as you can imagine, where does the whole notion of taking a  square of something or a square root come from? Well it comes from the notion of finding the  area of a square. If I have a square like this and if this side is seven, well if it's a square, all the  sides are going to be seven. And if I wanted to find the area of this, it would be seven times  seven or seven squared. That would be the area of this. Or if I were to say, well what is if I have a square, if I have, and that doesn't look like a perfect square, but you get the idea, all the sides  are the same length. If I have a square with area X. If the area here is X, what are the lengths of  the sides going to be? Well it's going to be square root of X. All of the sides are going to be the  square root of X, so it's going to be the square root of X by the square root of X and this side is  going to be the square root of X as well and that's going to be the square root of X as well. So  that's where the term square root comes from, where the square comes from. Now what do  you think cube root? Well same idea. If I have a cube. If I have a cube. Let me do my best  attempt at drawing a cube really fast. If I have a cube and a cube, all of it's dimensions have the  same length so this is a two, by two, by two cube, what's the volume over here. Well the 

volume is going to be two, times two, times two, which is two to the third power or two cubed.  This is two cubed. That's why they use the word cubed because this would be the volume of a  cube where each of its sides have length two and this of course is going to be equal to eight.  But what if we went the other way around? What if we started with the cube? What if we  started with this volume? What if we started with a cube's volume and let's say the volume here is eight cubic units, so volume is equal to eight and we wanted to find the lengths of the sides.  So we wanted to figure out what X is cause that's X, that's X, and that's X. It's a cube so all the  dimensions have the same length. Well there's two ways that we could express this. We could  say that X times X times X or X to the third power is equal to eight or we could use the cube root symbol, which is a radical with a little three in the right place. Or we could write that X is equal  to, it's going to look very similar to the square root. This would be the square root of eight, but  to make it clear, they were talking about the cube root of eight, we would write a little three  over there. In theory for square root, you could put a little two over here, but that'd be  redundant. If there's no number here, people just assume that it's the square root. But if you're  figuring out the cube root or sometimes you say the third root, well then you have to say, well  you have to put this little three right over here in this little notch in the radical symbol right  over here. And so this is saying X is going to be some number that if I cube it, I get eight. So  with that out of the way, let's do some examples. Let's say that I have... Let's say that I want to  calculate the cube root of 27. What's that going to be? Well if say that this is going to be equal  to X, this is equivalent to saying that X to the third or that 27 is equal to X to the third power. So  what is X going to be? Well X times X times X is equal to 27, well the number I can think of is  three, so we would say that X, let me scroll down a little bit, X is equal to three. Now let me ask  you a question. Can we write something like... Can we pick a new color? The cube root of, let  me write negative 64. I already talked about that if we're talking the square root, it's fairly  typical that hey you put a negative number in there at least until we learn about imaginary  numbers, we don't know what to do with it. But can we do something with this? Well if I cube  something, can I get a negative number? Sure. So if I say this is equal to X, this is the same  thing as saying that negative 64 is equal to X to the third power. Well what could X be? Well  what happens if you take negative four times negative four times negative four? Negative four  times four is positive 16, but then times negative four is negative 64 is equal to negative 64. So  what could X be here? Well X could be equal to negative four. X could be equal to negative four.  So based on the math that we know so far you actually can take the cube root of a negative  number. And just so you know, you don't have to stop there. You could take a fourth root and in  this case you'd have a four here, a fifth root, a sixth root, a seventh root of numbers and we'll  talk about that later in your mathematical career. But most of what you're going to see is  actually going to be square root and every now and then you're going to see a cube root. Now  you might be saying, well hey look, you know, you just knew that three to the third power is 27,  you took the cube root, you get X, is there any simple way to do this? And like you know if i give you an arbitrary number. If I were to just say, I don't know, if I were to say cube root of 125. And  the simple answer is, well the easiest way to actually figure this out is actually just to do a  factorization and particular prime factorization of this thing right over here and then you would figure it out. So you would say, okay well 125 is five times 25, which is five times five. Alright, so  this is the same thing as the cube root of five to the third power, which of course, is going to be 

equal to five. If you have a much larger number here, yes, there's no very simple way to  compute what a cube root or a fourth root or a fifth root might be and even square root can get quite difficult. There's no very simple way to just calculate it the way that you might multiply  things or divide it. 

5th roots 

Let's see if we can calculate the fifth root of 32. So, like always, pause the video and see if you  can figure this out on your own. So, let's just remind ourselves what a fifth root is. So, if x is  equal to the fifth root of 32, that's the same thing as saying x to the fifth power is equal to 32.  So, we have to find some number where, if you take five of them and multiply them together,  you'd get 32. So, there is a couple of ways to approach this. Especially when you're dealing with these really high order roots here. So, let me rewrite the fifth root of 32 here. One way is you  could try to factor 32 and see are there factors that show up five times? So, 32 you might  immediately recognize is an even number. So, it's gonna be divisible by two. It's two times 16.  16 is two times eight. Eight is two times four. Four is two times two. So, in this case, doing the  factoring technique worked out well. 'Cause we see that this is two times two times two times  two times two or two to the fifth power. You could rewrite this as the fifth root of two to the  fifth power, which is, of course, going to be equal to two. Two to the fifth power is 32. Now, let's do another one. It's gonna be a little bit harder. Let's say we wanna take the fifth root of 243.  So, now, a much, much larger number. So, there's a couple of ways to do this. One, you could  try the factoring. Although, that's gonna be harder now that it's a larger number. Or you could  do a little bit of trial and error. Doing higher roots without the aid of some type of calculator or  something is a little bit more complicated. So, here, if we wanted to do the factoring technique. We could say, alright, it's not divisible by two. I like to start with the smallest possible factor. So, it's not divisible by two. Is it divisible by three? And you might be familiar with the test to see if  something is divisible by three. You add up the digits and see if that sum of the digits is divisible by three. So, if I were to take two plus four plus three, that is equal to nine. And so it is divisible  by three. So, this is going to be equal to three times... Let's see three goes into 240 80 times  and then one. So, 81 times. And so, 81 is also divisible by three. I have a sense of where this is  going now. It's three times 27, which is three times nine. Which is three times three. So, using  the factoring method, we're able to see that three to the fifth power is 243. So, the fifth root of  243 is equal to three. Now, another way that you could have done it is a little bit of trial and  error. We already know.. Well, we know that one to the fifth power is just going to be one. We  know that two to the fifth power... We just calculated that. That's 32. Well, we now know what  three to the fifth is. Let's say we're just trying to zoom in on it a little bit. So, let's say, if you  wanted to see what four to the fifth is. Well, that would be four times four times four times four times four. So, let's see, this is going to be 16. 16 times four is 64. Times four is 256. And then,  that times four... And I just happen to know this. But you might wanna do it by hand. This is  1024. So, if you're taking the cube root of 243, you're saying what to the fifth power...  Something to the fifth power is equal to 243. And, if you have a sense that it's going to be an  integer solution, if you think it's going to be something like a two or a three, well, then, three is  probably going to be a good guess here. If the possible answers are gonna be decimals, then it's going to be a lot more complicated. But that's another way. Say, hey, maybe I'll try a three. 

And, if you try out three, you would get 243.