Fun with Exponents!

Using the properties of exponents, $\begin{aligned} \frac{x^{a^2}}{x^{b^2}} &= x^{32} \\ x^{a^2-b^2}&= x^{32} \\ a^2-b^2 &= 32 \\ (a-b)(a+b) &= 32 \\ 2(a+b) &= 32 \\ a+b &= 16 \end{aligned}$

Since $a-b=2$ and $a+b = 16$ , the values of $a$ and $b$ are $9$ and $7$ , respectively. Thus $ab=\boxed{63}$ .