Irrationality of powers of $\pi$

Question

Everyone knows that $\pi$ is an irrational number, and one can refer to this page http://planetmath.org/encyclopedia/PiAndPi2AreIrrational.html for the proof that $\pi^{2}$ is also irrational.

What about the Highers powers of $\pi$, meaning is $\pi^{n}$ irrational for all $n \in \mathbb{N}$ or does there exists a $m \in \mathbb{N}$ when $\pi^{m}$ is rational.

Answer 1

What Robin hinted at:

If $\pi^{n}$ was rational, then $\pi$ would not be transcendental, as it would be the root of $ax^{n}-b = 0$ for some integers $a,b$.