How would one convert this problem into an equation that could then be simplified?
A right-angled triangle has shorter side lengths exactly $a^2-b^2$ and $2ab$ units respectively, where a and b are positive real numbers such that a is greater than b. Find an exact expression for the length of the hypotenuse (in appropriate units).
So obviously I would plug those values into the Pythagorean Theorem.
$c = \sqrt{(a^2 – b^2)^2 + (2ab)^2}$
So then I try to simplify it but I always have the $\sqrt{2}$ left over which isn't in any of the multiple choice questions on MathXL. I have been stuck on this problem for 30 minutes. Any help would be much appreciated.
Best Answer
Things are set up nicely to factor as squares; in particular,
$$(a^2 - b^2)^2 + (2ab)^2 = a^4 + b^4 - 2a^2 b^2 + 4a^2 b^2 = a^4 + 2a^2b^2 + b^4 = (a^2 + b^2)^2.$$
This should simplify things reasonably.