Which one is greater - $e^{\pi}$ or $\pi^{e}$?

Recall that: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$

Therefore, $\forall x > 0$, $e^x > 1 + x$.

Substituting $x = \frac{\pi}{e} - 1$ in the above inequality, we get:

$e^{\frac{\pi}{e} - 1} > \frac{\pi}{e}  \implies e^{\frac{\pi}{e}} > \pi  \implies e^{\pi} > \pi^{e}$