Existence of Rational Powers of Irrational Numbers/Proof 1
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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.
$\blacksquare$