Rationality of a power with irrational exponent The following fact is known:
If $a$ and $b$ are both irrational numbers, then $a^b$ can be a rational number.
Proof. Suppose $a^b$ is always irrational. Then $\sqrt{2}^\sqrt{2}$ is an irrational number, which in turn implies that  $\left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2}$ must also be irrational. However:
$$
\left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2} = \sqrt{2}^{\sqrt{2}\times\sqrt{2}} = 2
$$

Now assume $a$ is a rational number and $b$ is irrational. I wonder if  $a^b$ can be a rational number. Any way to prove, or a reference maybe?

$2^b = 3$ where $b = \log_2(3)$ is irrational.  In fact, if $2^b = c$ where $b$ and $c$ are rational, $b$ must be an integer.
