In 1934, Gelfond and Schneider independently proved the following theorem, which now bears their names:

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.)