can I get some help for proving that $z^{-w} = \frac{1}{z^w}$ where $z, w \in \mathbb{C}$ for any branch of the logarithm
sorry for the formation not sure how to write math types here yet
2026-04-28 20:04:49.1777406689
Negative complex number in exponent
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Let $z^{-w}=k$
Then taking ln on both sides,
$ln(z^{-w})=-wlnz=ln k$
Since $-wlnz=wln\frac{1}{z}=lnk$
We get $k=\frac{1}{z}^w$