# How do I simplify this surd expression?

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.

• You should add a term $ab$ and make it an absurd expression. – user296602 Dec 10 '18 at 17:53
• What do you get when you multiply out everything under the radical (surd)? Can you factor it? – Doug M Dec 10 '18 at 17:58

$$(a^2 - b^2)^2 + (2ab)^2 = a^4 + b^4 - 2a^2 b^2 + 4a^2 b^2 = a^4 + 2a^2b^2 + b^4 = (a^2 + b^2)^2.$$
Hint: How can $$x^2+2xy+y^2$$ be factored?