TITLE: Why can't I combine complex powers
QUESTION [17 upvotes]: I came across this 'paradox' -
$$1=e^{2\pi i}\Rightarrow 1=(e^{2\pi i})^{2\pi i}=e^{2\pi i \cdot 2\pi i}=e^{-4\pi^2}$$
I realized the fallacy lies in the fact that in general $(x^y)^z\ne x^{yz}$. Why doesn't it work with complex numbers even though it is valid in real case? Is it related to the fact that logarithm of complex number is not unique?

REPLY [5 votes]: This is related to a note by Euler, maybe he was the first to realize that $i^i$ is real. Actually, 
$$i^i = (e^{i\pi/2})^i = e^{-\pi/2}$$
on the other hand 
$$i^i = (e^{i(\pi/2+2\pi n})^i = e^{-\pi/2 -2\pi n},\ n\in\mathbb{Z}.$$
So maybe it is better to say that $i^i$ is a subset of the $\mathbb{R}$ and that certain equality signs are to be understood as congruences.