## An algebraic property of complex numbers

I'm guessing that if $z\in \mathbb C$, then we have

$$\left| z^{-1/2} \right|^2 = |z^{-1}| = |z|^{-1} = \frac{1}{|z|}.$$

Proving this seems to be a real headache though. Is there a quick/easy way to do this?
 write z in polar form?
 Recognitions: Science Advisor Don't mean to nitpick, but remember that it is for z in ℂ\{0} to start with; some profs. may take away points in an exam if you don't specify this. But also, remember your square root is not defined everywhere, at least not as a function, but as a multifunction, since every complex number has two square roots. I mean, the expression z1/2 is ambiguous until you choose a branch. Sorry if you already are taking this into account; I am in nitpicking mode, but I shouldn't take it out on you :) .

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