# Existence of Rational Powers of Irrational Numbers

## Theorem

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

## Proof 1

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.

$\blacksquare$

## Proof 2

Given that $2$ is rational and $\sqrt 2$ is irrational, consider the number $q = \sqrt 2^{\sqrt 2}$.

We consider the two cases.

- $(1): \quad$ If $q$ is rational then $a = \sqrt 2$ and $b = \sqrt 2$ are the desired irrational numbers.
- $(2): \quad$ If $q$ is irrational then $q^{\sqrt 2} = \paren {\sqrt 2^{\sqrt 2} }^{\sqrt 2} = \sqrt 2^{\paren {\sqrt 2} \paren {\sqrt 2} } = \sqrt 2^2 = 2$ is rational, so $a = q = \sqrt2^{\sqrt 2}$ and $b = \sqrt 2$ are the desired irrational numbers.

$\blacksquare$

## Proof 3

Consider the equation:

- $\left({\sqrt 2}\right)^{\log_{\sqrt 2} 3} = 3$

We have that $\sqrt 2$ is irrational and $3$ is (trivially) rational.

It remains to be proved $\log_{\sqrt 2} 3$ is irrational.

We have:

\(\ds \log_{\sqrt 2} 3\) | \(=\) | \(\ds \frac{\log_2 3}{\log_2 \sqrt 2}\) | Change of Base of Logarithm | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac{\log_2 3}{1/2}\) | Logarithms of Powers: $\sqrt 2 = 2^{1/2}$ and $\log_2 2 = 1$ | |||||||||||

\(\ds \) | \(=\) | \(\ds 2 \, \log_2 3\) |

From Irrationality of Logarithm, $\log_2 3$ is irrational.

Therefore $2 \, \log_2 3$ is irrational.

Thus there exist two irrational numbers $a = \sqrt 2$ and $b = \log_{\sqrt 2} 3$ such that $a^b$ is rational.

$\blacksquare$