# complex numbers such that $z^\alpha w^\alpha$ and $(zw)^\alpha)$ have different principal values?

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?

Sure -- for eample, $z=w=-1$ and $\alpha=1/2$.
(This gives rise to an often-seen fake proof that $-1=1$).
Sure, take $\alpha=1/2$ and let $z=w$ lie in the second quadrant (so that their product lies in the third).