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

$\blacksquare$

Proof 3

Consider the equation:

$\paren {\sqrt 2}^{\log_{\sqrt 2} 3} = 3$

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.

$\blacksquare$