Value of $2^{\sqrt{2}}$

Question

Can you evaluate the value of $2^{\sqrt{2}}$

I am not sure, if there is some trick to evaluate its value because it was asked in an interview.

But can we atleast comment if it is rational or irrational.

Obviously a rational raised to the power of irrational, can be either rational or an irrational number. So unless we do something we cannot simply comment about it nature. I am not getting any idea to proceed futher.

Answer 1

There isn't, no, you just need to use a calculator. It's an interesting number, but not one you can really write in another way.

By the Gelfond–Schneider theorem, it's a transcendental number.

It's also famously used in a much less advanced proof that some irrational $a,\,b$ satisfy $a^b\in\Bbb Q$. Take either $a=\sqrt{2},\,b=2\sqrt{2}$ if $2^\sqrt{2}$ is rational, or $a=2^\sqrt{2},\,b=\sqrt{2}$ if it's not.

(Actually, we usually use $b=\sqrt{2}$ in the first case or $a=\sqrt{2}^\sqrt{2}$ in the second, but it's a similar argument either way.)