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.