When does the reciprocal of a square root equal the square root of a reciprocal?

Question

By reciprocal is meant multiplicative inverse, i.e. given $x$, the reciprocal will be $1/x$, for this question. Why does the following fail in $\mathbb C$? (We require $x$ be a purely real variable, i.e. $\Im (x)=0$) $$ \frac{1}{\sqrt{x}} = \sqrt{\frac{1}{x}} $$ A simple counterexample is: $$ 1/i = \frac{1}{\sqrt{-1}} \neq \sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}} = i $$ Context: I came to the wrong result when doing some calculations because the denominator was negative, and I realized I hadn't been careful enough when combining square roots. A similar statement is: $$ \frac{\sqrt{a}}{\sqrt{b}} \neq \sqrt{\frac{a}{b}} $$ when $b<0$. I was curious about why this is invalid? There is no formal system I'm assuming, so that part of the question is open-ended (i.e. which axioms of the complex field are used).