How can I express $ i^{2i}$ in the form $x + iy$?

Question

I'm not sure how to begin since this is not in the form $re^{i \theta}$.

Answer 1

Write $i=e^{i\pi/2}$ then $i^{2i}=e^{2i^2\pi/2}=e^{-\pi}$

Note that $e^{i\theta}$ representations are only defined up to mutliples of $2\pi$ in $\theta$, so really this is only one of a myriad of answers, as

$$i=e^{2\pi i + i\pi/2}=i\implies i^{2i}=e^{4\pi i^2 +2i^2\pi/2} = e^{-5\pi}$$

and so on. So the right answer is that this is not even well-defined.