Existence of Rational Powers of Irrational Numbers/Proof 1

Theorem

There exist irrational numbers $a$ and $b$ such that $a^b$ is rational.

Proof

We have that:

$\sqrt 2$ is irrational.

$2$ is trivially rational, as $2 = \dfrac 2 1$.

Consider the number $q = \sqrt 2^{\sqrt 2}$, which is irrational by the Gelfond-Schneider Theorem.

Thus:

$q^{\sqrt 2} = \left({\sqrt 2^{\sqrt 2}}\right)^{\sqrt 2} = \sqrt 2 ^{\left({\sqrt 2}\right) \left({\sqrt 2}\right)} = \sqrt 2^2 = 2$

is rational.

So $a = q = \sqrt 2^{\sqrt 2}$ and $b = \sqrt 2$ are the desired irrationals.

$\blacksquare$