Why is this proof false? (Why is $e^i \neq 1$?)

Question

I found this on MathOverflow:

$$e^i = (e^i)^{(2\pi/2\pi)} = (e^{2\pi i})^{1/2\pi} = 1^{1/2\pi} = > 1.$$

I first saw this one many years ago, written on the wall of a bathroom stall in the Princeton University math department.

I don't know a whole lot about Euler's identity and related topics. Why is this proof false?