Answer
$x=\pm 2$ or $x=\pm 2i$
Work Step by Step
$ x^{4}=16\qquad$...substitute $x^{2}$ for $u$ so that $x^{4}=u^{2}$.
$u^{2}=16$
According to the general principle of square roots:
For any real number $k$ and any algebraic expression $x$ :
$\text{If }x^{2}=k,\text{ then }x=\sqrt{k}\text{ or }x=-\sqrt{k}$.
$u=\pm 4$
Bring back $x^{2}=u$
$x^{2}=4$ or $x^{2}=-4$
$x=\pm 2$ or $x=\pm 2i$