Simplifying square-root expressions: no variables (advanced)

- [Voiceover] We're asked to simplify the expression by removing all factors that are perfect squares from inside the radicals and combining the terms. So, let's see if we can do it and pause the video and give a-go at it before we do it together. Alright, so let's see how we can re-write these radicals. So, four times the square root of 20. Well, thats the same thing as four times the square root of four, times the square root of five, because 20 is the same thing as four times five. And 45, that's the same thing as nine times five. And the reason why I'm thinking about the four and I'm thinking about the nine is because those are perfect squares, so I could write this as, four times the square root of four times the square root of five. And then I could say, this part right over here is minus three, times the square root of nine, times the square root of five. The square root of 45 is the same thing as the square root of nine times five, which is the same thing as square root of nine times the square root of five, and then all of that is going to be over, all of that is over the square root of 35. Now, are there any perfect squares in 35? 35 is seven times five. No, neither of those are perfect squares. So, I could just leave that as square root of 35. And, let's see, the square root of four? Well, thats going to be two. This is the principal root, so we're thinking about the positive square root. The square root of nine is three. And so, this part right over is going to be four times two, times the square root of five, so it's going to be eight square roots of five. And then, this part over here is going to be minus three, times three, times the square root of five. So, minus nine square roots of five. And all of that is going to be over the square root of 35. Square root of 35. And, so let's see, if I have eight of something, and I subtract nine of that something, I'm gonna have negative one of that something. So, I could say negative one times the square root of five, or I could just say negative square root of five. Negative square root of five over the square root of 35. I actually think I could simplify this even more, because this is the same thing. This is equal to the negative of the square root of five over 35. Both the numerator and the denominator are divisible by five. So, we could divide them both by five, and we would get the square root of divide the numerator by five, you get one. Divide the denominator by five, you get seven. So, we can view this as the square root of 1/7th. Square root of 1/7th. And we are all done. Let's do another one of these. These are strangely, strangely fun. And once again, pause it, and see if you can work it out on your own. Perform the indicated operations. Alright, so let's first multiply. So, this essentially is doing the distributive property twice. And, actually let me just do it that way, so let's distribute the square root of five, plus the square root of six. Let's first multiply it times the square root of five. So, the square root of five times the square root of five is going to be five. Square root of five times the square root of six is the square root of 30. So, five plus the square root of 30. And then, when I take this expression, and I multiply it times the second term, times the negative square root of six. Well, negative square root of six times the square root of five is going to be the negative of the square root of 30. And then the negative of the square root of six times the square root of six is going to be, we're gonna subtract six. The square root of six times the square root of six is six, then we have the negative out there. And so, just like that, we are left with, well, let's see. Square root of 30 minus the square root of 30, well those cancel out, thats zero. And we're left with five minus six, which is going to be equal to negative one. And we are, we're, we're all done. Now, another way that you could've viewed this, is you could've seen a pattern here. You could've said, Well, this is the same thing as A plus B, times A minus B, or A is square root of five and B is square root of six, and we know that this will result in the difference of squares. This is will be A squared minus B squared. And, so this for this particular case it would be square root of five squared, minus square root of six squared. Which is of course equal to five minus six, which is equal to negative one. Either way, hopefully you found that vaguely entertaining.