We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this:
Take $(\sqrt{2})^{\sqrt{2}}$
If it's rational, then you proved it, if it's irrational, take $((\sqrt{2})^{\sqrt{2}} ){^\sqrt{2}} = 2$ and you've proved it.
I'm wondering if you can raise $\pi$ or $e$ to a certain non-trivial real power to make it rational? And if not, where is the proof that it can't be done?
p.s. - I almost left out the real part, but then I realized that $e^{i\pi} = -1$.
Of course. Pick any positive rational $p$ and let $x=\log_\pi p$, then $\pi^x=p$.