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# Simplifying square-root expressions: no variables

## Video transcript

let's get some practice rewriting and simplifying radical expressions so in this first exercise and these are all from Khan Academy it says simplified the expression by removing all factors that are perfect squares from inside the radicals and combine the and combine the terms if the expression cannot be simplified enter it as given all right let's see what we can do here so we have negative 40 okay the negative square root of 40 I should say we write a little bit bigger so you can see that so the negative square root of 40 plus the square root of 90 so let's see what perfect squares are in for T so what immediately jumps out at me that this it's divisible by 4 and 4 is a perfect square so this is the negative square root of 4 times 10 plus the square root of well what jumps out at me is this is divisible by 9 9 is a perfect square so 9 times 10 and if we look at the tens here 10 10 does not have any perfect squares in it anymore if you wanted to do a full factorization of 10 a full prime factorization it would be it would be 2 times 5 so there's no perfect squares in 10 and so we can work it work it out from here this is the same thing as the negative of the square root of 4 times the square root of 10 plus the square root of 9 times the square root of 10 and we take when I say square root I'm really saying principal root the positive square root so it's the negative of the positive square root of 4 so that is so let me do this in another color so it can be clear so this right here is 2 this right here is 3 so it's going to be equal to negative 2 square roots of 10 plus 3 square roots of 10 so if I have negative 2 of something and I add 3 of that same something to it that's going to be what well that's going to be 1 square root of 10 now this last step doesn't make full sense actually let me let me slow it down a little bit I could I could rewrite it this way I could write it as 3 square roots of 10 minus 2 square roots of 10 that might jump out it a little bit clearer if have three of something and I were to take away two of that something and that's case of square roots of tens well I'm going to be left with just one of that something I'm just going to be left with one square root of 10 which we could just write as the square root of 10 another way to think about it is another way to think about it is we could factor out a square root of 10 here so you undistribute it do the distribute distributive property in Reverse that would be the square root of 10 times 3 minus 2 which is of course this is just 1 so you're just left with the square root of 10 so all of this simplifies to square root of 10 let's do a few more of these so this says simplify the expression by removing all factors that are perfect squares from inside the radicals and combining the terms so essentially the same same same idea alright let's see what we can do so this is interesting we have a square root the square root of 1/2 so can I well actually what could be interesting is if I have a square root of something times the square root of something else so the square root of 180 times the square root of 1/2 this is the same thing as the square root of 180 times 1/2 and this just comes straight out of our exponent properties it might look a little bit more familiar if I wrote it as 180 to the 1/2 power times 1/2 to the 1/2 power is going to be equal to 180 times 1/2 to the one half power taking the square root the principal root is the same thing as raising something to the one-half power and so this is the square root of 80 times 1/2 which is going to be the square root of 90 which is equal to the square root of 9 times 10 and we just we just simplified square root of 90 in the last problem that's equal to the square root of 9 times the square root of the principal root of 10 which is equal to 3 times the square root of 10 3 times the square root of 10 all right let's keep going so I have one more of these examples and like always pause the video and see if you can work through these on your own before I work it out with you simplify the expression by removing all factors that are perfect squares okay these are the same directions that we've seen the last few times and so let's see if I wanted to do if I wanted to simplify this this is equal to the square root of well 64 times 2 is 128 and 64 is a perfect square so I'm gonna write it as 64 times 2 over 27 is 9 times 3 9 is a perfect square so this is going to be the same thing and there's a couple of ways that we could think about it we could say this is the same thing as the square root of 64 times 2 over the square root of 9 times 3 which is the same thing as the square root of 64 times the square root of 2 over square root of 9 times the square root of 3 which is equal to this is 8 this is 3 so it would be 8 times the square root of 2 over 3 times the square root of 3 that's one way to say it or we could even view the square root of 2 over the square root of 3 as a square root of 2/3 so we could say this is 8 over 3 times the square root of 2/3 so these are all possible ways of trying to tackle this so we could just write it let's see have we removed all factors that are perfect squares yes from inside the radicals and we've combined terms there we weren't doing any adding or subtracting here so it's really just removing the perfect squares from inside the radicals and I think we've done that so we could say this is going to be 8/3 times the square root of 2/3 and there's other ways that you could express this that would be equivalent but hopefully this this makes some sense