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CCSS.Math:

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 can 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 not just 49 if I write this 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 principal root now typically and as you advanced 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 non-negative this is going to be this is going to change once we start thinking about thinking about imaginary and complex numbers but typically for the principal square root we assume that whatever is under the radical whatever is 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 this where does this where's the whole notion of taking a square of something or 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 7 well if it's a square all the sides are going to be 7 and if I wanted to find the area of this it would be 7 times 7 or 7 squared that would be the area of this now 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 the sides are going to be square root of x so it's going to be square root of x by square root of x and this side is going to be square root of X as well and that's going to be 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 I mean my best attempt at drawing a cube really fast if I have a cube and a cube all of its dimensions have the same length so this is a 2 by 2 by 2 cube what's the volume over here well the volume is going to be 2 times 2 times 2 which is 2 to the third power or 2 cubed this is 2 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 2 and this of course is going to be equal to 8 but what if we went the other way around what if we started with the cube and what if we started with its volume what if we if we started with a cubes volume and let's say the volume here is 8 cubic units so volume is equal to 8 and we wanted to find the lengths of the sides so we wanted to figure out what X we wanted to figure out what X is because that's X that's X and that's X it's it's a cube so all this 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 8 or we could use the cube root symbol which is a radical with a little 3 in the right place or 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 8 but to make it clear that were talking about the cube root of 8 we would write a little 3 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 to figure out the the cube root notice sometimes you say the third root then well then you have to say well you have to put this little three right over here in the 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 on the way let's let's do some examples let's say that I have let's say that I want to calculate the cube root the cube root of 27 what's that going to be well if I 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 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 multiply if I cube something can I get a negative number sure if I take 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 4 times negative 4 times negative 4 negative 4 times negative 4 is positive 16 but then times negative 4 is negative 64 is equal to negative 64 so what could X be here well X could be equal to negative 4 X could be equal to negative 4 so based on the map 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 4 in case you would have a four-year a fifth through to six through to seventh root of numbers and we'll talk about that in well later in your mathematical career but most of what you 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 cube root of 125 and the simple answer is well you know the easiest way to actually figure this out is actually just to do a factorization in particular private of this thing right over here and then you would figure it out so you would say okay well 125 is 5 times 25 which is 5 times 5 all right so this is the same thing as the cube root of 5 to the third power which of course is going to be equal to which is going to be equal to 5 if you have a much larger number here yes it there's no there's no very simple way to compute what a cube root or a fourth root or fifth root might be an even square root can get quite difficult it doesn't there's no very simple way to just calculate it the way that you might multiply things or divide it