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.

The problem is that $(e^x)^y=e^{xy}$ does not hold with complex numbers as it does with real numbers.  This can change what the principal value, which is what has happened in your example.  You should read a bit about principal logarithms and branch cuts.
