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# Simplifying square roots (variables)

CCSS.Math:

## Video transcript

what I want to do in this video is Reis implement retime z-- the principal root of 500 times X to the third and take it into consideration some of the comments that we got out on YouTube that actually gives some interest interesting perspective on how you can simplify this so just as a quick review of what we did in the last video we said that this is the same thing as 3 times the principal root of 500 and I'm going to do it a little bit different than I did in the last video just just to make it interesting this is 3 times the principal root of 500 times the principal root of x to the third and 500 we can rewrite it because 500 is not a perfect square we can rewrite 500 as 100 times 5 or even better we can rewrite that as 10 squared times 5 10 squared is the same thing as 100 so we can rewrite this first part over here as 3 times the principal root of 10 squared times 5 times x times the principal root of x squared times X that's the same thing as X to the third now the one thing I'm going to do here actually I won't talk about it just yet of how we're going to do it differently than we just in the last video this radical right here can be rewritten as so this is going to be 3 times the square root or the principal root I should say of 10 squared times the square root of 5 if we take the square root of the product of two things it's the same thing as taking the square root of each of them and then taking the product and so and this over here is going to be times the square root of or the principal root of x squared times the principal root of x and the principal root of 10 squared is 10 and then what I said in the last video is that the principal root of x squared is going to be the absolute value of x just in case just in case X itself is a negative number and so then if you simplify all of this you get 3 times 10 which is 30 times and I'm just going to switch the order here times the absolute value of x and then you have the square root of 5 or the principal root of 5 times the principal root of X and this is just going to be equal to the principal root of 5 X taking the square root of something and multiplying that times the square root of something else is the same thing just taking the square root of 5x so all of this simplified down to 30 times the absolute value of x times the principal root of 5x and this is what we got in the last video and the interesting thing here is if we assume we're only dealing with real numbers the domain of X right over here the X's that will make this expression defined in the real numbers then X has to be X has to be greater than or equal to 0 so let me so maybe I could write it this way the domain here the domain here is that X is X any any real number real number greater than or equal to 0 and the reason why I say that is if you put a negative number in here if you put a negative number in here and you cube it you're going to get another negative number and then it doesn't make it sit at least in the real numbers it that you won't get you won't get an actual you won't get an actual value you'll get a square root of a negative number here so if you make this if you assume this right here we're dealing with the real numbers we're not dealing with any complex numbers when you when you open it up the complex numbers then you can have a then you can't expand the domain more broadly but if you're dealing with real numbers you can say that X is going to be greater than or equal to 0 and then the absolute value of x is just going to be X because it's not going to be a negative number and so if we're assuming that the domain of X is or if this bout this expression is going to be valuable or it's or it's going to have a positive number then this can be written as 30 x times the square root of 5 X if you if you if you had the situation where we were dealing with complex numbers then you would so numbers that where and if you don't want a complex number is or an imaginary number don't worry too much about it but if you were dealing with those then you would have to keep the absolute value of x there because then this would be defined this would be defined for numbers that are less than 0