Show that $e^{\sqrt 2}$ is irrational

Since the sum of two rational numbers is rational, one or both of $e^{\sqrt{2}}$ and $e^{-\sqrt{2}}$ is irrational. But, $e^{-\sqrt{2}}=1/e^{\sqrt{2}}$, and hence both are irrational.

$e^{\sqrt{2}}$ is transcendental because of Lindemann–Weierstrass theorem:

If $a\neq 0$ is algebraic, then $e^a$ is transcendental.

It is written in the list of transcendental numbers.