$\displaystyle \sqrt{9^{16x^2}}=\begin{cases}\text{a) }3^{4x} &\text{b) } 3^{16x^2}\\\text{c) }9^{4x^2}&\text{d) }9^{8x}\end{cases}$
$\displaystyle \sqrt{9^{16x^2}} \ = \ \sqrt{(3^2)^{16x^2}} \ = \ \sqrt{3^{[2(16x^2)]}} \ = \ \sqrt{(3^{16x^2})^2} \ = \ 3^{16x^2} $ **
** $\displaystyle \sqrt{A^2} \ = \ |A|, $ but the expression for A here is always non-negative, so the final expression in the above line follows.