Rationality test!

Take this example: $\sqrt{2}^{\sqrt{2}}$ .

If $\sqrt{2}^{\sqrt{2}}$ is a rational number, then the problem is solved.

If $\sqrt{2}^{\sqrt{2}}$ is an irrational number, then consider this case:

• $(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^{\sqrt{2} \times \sqrt{2}} = \sqrt{2}^2 = 2$ (2 is, obviously, a rational number)

Therefore, there is always a case where an irrational number raised to the power of another irrational number results in a rational number.

After studying a little bit of log, I thought this would be a good problem.

$a^{\log_{a}{b}} =b$ , putting any irrational number as $a$ and any rational number as $b \neq a^2$ , we have infinite answers!

P.S. I need practice for log, as I just studied it today, so I may be wrong. Please comment if that is the case.