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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).

Sam Gallagher
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    There are many questions on this site about square root ambiguities. Bottom line is that for nonnegative reals the square root symbol always means the nonnegative root. For negative reals or complex numbers there's no uniform way to choose which root that preserves all the rules for exponents. – Ethan Bolker Dec 01 '22 at 20:34
  • @EthanBolker Feel free to mark this as a duplicate if you have an example of the same question asked somewhere else. – Sam Gallagher Dec 01 '22 at 20:45
  • @Mirko I do like that better, changing it now. – Sam Gallagher Dec 01 '22 at 20:45

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