GelfondSchneider Theorem 

Suppose $a$ and $b$ are algebraic numbers. Then $a^b$ is transcendental unless a=0, a=1, or b is rational.
(Obviously, if any of those conditions hold then $a^b$ is in fact algebraic.)
This theorem implies that, for instance, $sqrt{2}^sqrt{2}$ and $e^\pi$ are transcendental. (The latter because otherwise $1=e^{i\pi}=(e^\pi)^^i$ would be transcendental.)