Question about complex numbers (what's wrong with my reasoning)? Can someone point out the flaw here?
$$e^{-3\pi i/4} = e^{5\pi i/4}$$
So raising to  $\frac{1}{2}$, we should get
$$e^{-3\pi i/8} = e^{5\pi i/8}$$
but this is false.

Paraphrase using $e^0=1$ and $e^{\pi i}=-1$. We can write
$$
e^{-3\pi i/4}\;1^2=e^{-3\pi i/4}\;(-1)^2
$$
Raising to the $\frac12$ power yields
$$
e^{-3\pi i/8}\;1=e^{-3\pi i/8}\;(-1)
$$
The problem is that without proper restrictions (e.g. branch cuts), the square root is not well-defined on $\mathbb{C}$.
