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Is it possible to find two complex numbers $w,z$ and a complex exponents $\alpha$ such that the principal values of $z^\alpha w^\alpha$ and $(zw)^\alpha)$ are different?

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Sure -- for eample, $z=w=-1$ and $\alpha=1/2$.

(This gives rise to an often-seen fake proof that $-1=1$).

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Sure, take $\alpha=1/2$ and let $z=w$ lie in the second quadrant (so that their product lies in the third).

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