What's wrong with my simplification of $\frac1i$?

Question

Supposedly, $\frac1i=-i$. However, my simplification reaches another result: $$\frac1i=\frac{\sqrt1}{\sqrt{-1}}=\sqrt{\frac1{-1}}=\sqrt{-1}=i\text.$$ What's wrong with my process?

Answer 1

The formula $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ is only guaranteed to be valid when $a,b$ are positive real numbers.

For complex numbers, all bets are off.