#### Answer

$\color{blue}{\left\{-9i, 9i\right\}}$

#### Work Step by Step

RECALL:
If $x^2=a$, then taking the square root of both sides gives $x = \pm \sqrt{a}$.
Take the square root of both sides of the given equation to obtain:
$\sqrt{x^2}=\pm \sqrt{-81}
\\x =\pm \sqrt{81(-1)}
\\x =\pm 9\sqrt{-1}$
Since $\sqrt{-1}=i$, then the solution is:
$x=\pm 9i$
Thus, the solution set is $\color{blue}{\left\{-9i, 9i\right\}}$.