Simplifying more involved radical expressions

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. Pause the video and give it a go at it before we do it together.

All right, so let's see how we can rewrite these radicals. So, 4 times the square root of 20, well, that's the same thing as 4 times the square root of 4 times the square root of 5 because 20 is the same thing as 4 times 5.

And 45, and that's the same thing as 9 times 5. The reason why I'm thinking about the 4 and I'm thinking about the 9 is because those are perfect squares. So I could write this as 4 times the square root of 4 times the square root of 5.

Then I could say this part right over here is minus 3 times the square root of 9 times the square root of 5. The square root of 45 is the same thing as the square root of 9 times 5, which is the same thing as the square root of 9 times the square root of 5.

And then all of that is going to be over the square root of 35. Now, are there any perfect squares in 35? 35 is 7 times 5. No, neither of those are perfect squares, so I could just leave that as the square root of 35.

Let's see, the square root of 4, well, that's going to be 2. This is the principal root; we're thinking about the positive square root. The square root of 9 is 3, and so this part right over here is going to be 4 times 2 times the square root of 5.

So, it's going to be 8 square roots of 5, and then this part over here is going to be minus 3 times 3 times the square root of 5. So, minus 9 square roots of 5.

All of that is going to be over the square root of 35. So, let's see if I have 8 of something and I subtract 9 of that something, I'm going to have negative 1 of that something.

So, I could say negative 1 times the square root of 5, or I could just say negative square root of 5 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 5 over 35.

Both the numerator and denominator are divisible by 5, so we could divide them both by 5, and we would get the square root of—divide the numerator by 5, you get 1, divide the denominator by 5, you get 7. So we could view this as the square root of one-seventh.

Square root of one-seventh, and we are all done. Let's do another one of these; these are strangely fun. Once again, pause it and see if you can work it out on your own. Perform the indicated operations.

All right, 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, square root of five times square root of five is going to be 5. Square root of five times the square root of six is the square root of 30. So, 5 plus the square root of 30.

Then when I take this expression and I multiply it times the second term, times the negative square root of 6, well, negative square root of 6 times the square root of 5 is going to be the negative of the square root of 30.

Then the negative of the square root of 6 times the square root of 6 is going to be—we're going to subtract 6. The square root of 6 times the square root of 6 is 6, and we have the negative out there.

So just like that, we are left with—well, let's see. Square root of 30 minus square root of 30, well, those cancel out; that's 0, and we're left with 5 minus 6, which is going to be equal to negative 1.

And we are all done. Now, another way that you could have viewed this is you could have seen a pattern here. You could have said, well, this is the same thing as a plus b times a minus b, where a is square root of 5 and b is square root of 6.

We know that this will result in the difference of squares. This will be a squared minus b squared, and so for this particular case, it would be square root of 5 squared minus square root of 6 squared, which of course is equal to 5 minus 6, which is equal to negative 1.

Either way, hopefully you found that vaguely entertaining.