How is $\frac{1}{\sqrt{-16}} = -\frac{1}{4}$?

Question

Simplify $(-16)^{ -\frac{1}{2}}$

Here is my attempt at solving this question:

\begin{align} (-16)^{ -\frac{1}{2} } &= \frac{1}{(-16)^\frac{1}{2}} \\ &= \frac{1}{\sqrt{-16}} \end{align}

$$ \therefore \text{The answer is undefined} $$

This was the answer:

\begin{align} (-16)^{ -\frac{1}{2} } &= \left( \frac{1}{-16} \right) ^\frac{1}{2} \\ &= \frac{1}{\sqrt{-16}} \\ &= -\frac{1}{4} \end{align}

Isn't $\sqrt{-16}$ invalid? How can $\frac{1}{\sqrt{-16}}$ be simplified to $-\frac{1}{4}$?

Answer 1

If you disallow imaginary numbers, it is undefined. Otherwise, there are two solutions to $x^2=-1/16$, namely $$\pm\frac{1}{4i}=\pm\frac{i}{4},$$ where $i=\sqrt{-1}$. In any case, it is not $-1/4$.