Intro to cube roots

- [Voiceover] 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.