Strange question about exponents I've been stumped for a while trying to figure out the flaw in this logic. Consider some positive, real x. Then 
$$e^{ix} = e^{2\pi i\frac{x}{2\pi}} = (e^{2\pi i})^{\frac{x}{2\pi}} = 1^{\frac{x}{2\pi}} = 1$$
I've felt like an idiot for hours. What am I missing?

$z \to z^{\frac x {2\pi}}$ is a mutivalued function and just one of its values for $z=e^{2\pi i}$  is $e^{2\pi i\frac x {2\pi}}$. So you cannot conclude that $e^{ix}=1$.
To understand exactly what is happening you have to get familiar with logarithms in the complex plane. 
