Video: Simplifying Expressions That Include Radicals and Exponents

Simplify (√(15) + √(19))²(√(15) − √(19))²

02:27

Video Transcript

Simplify the square root of 15 plus the square root of 19 all squared times the square root of 15 minus the square root of 19 all squared.

Now, if you’ve been paying attention throughout the video, you’re probably spotting the fact, “Oh, that looks like difference of two squares!” But it’s not quite in that format, is it? So the difference of two squares says that 𝑎 squared minus 𝑏 squared is 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And we’re nearly there. But we’ve got this problem of these two squareds above the parentheses here. So if we do a bit of rearranging then, we will be able to use that. So let’s have a look at it. Well, root 15 plus root 19 all squared is just root 15 plus root 19 times root 15 plus root 19. So that’s the first parentheses dealt with, and similarly for the second.

And now, we’ve got four sets of parentheses, all multiplied together. And when we multiply things together, it doesn’t really matter what order we do them in. So I’m gonna rearrange these. So I’ve just swapped around the order of the middle two sets of parentheses there. And that’s left me with root 15 plus root 19 times root 15 minus root 19 all times root 15 plus root 19 times root 15 minus root 19. Now each of those is the difference of two squares format that we were looking for. So if 𝑎 is root 15 and 𝑏 is root 19, we have this pattern here. And we also have it here. And that means that 𝑎 squared is root 15 all squared, which is 15. And 𝑏 squared is root 19 all squared, which is 19. And 𝑎 squared minus 𝑏 squared is 15 minus 19.

So let’s go back to our question then. We’ve got 𝑎 plus 𝑏 times 𝑎 minus 𝑏 times 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And 𝑎 plus 𝑏 times 𝑎 minus 𝑏 is the same as 𝑎 squared minus 𝑏 squared. And as we just said, 𝑎 squared minus 𝑏 squared is 15 minus 19. And 15 minus 19 is negative four. So that becomes negative four times negative four which is positive 16.

So remembering our difference of two squares and doing a little bit of reorganising to get things in the right format for that has saved us an awful lot of multiplying out. And got us to a very simple answer of 16 instead of all of this stuff up here, which we started off with at the beginning of the question.

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