If $q^n$ is irrational for all $n>1$, then $q$ is irrational. Theorem. Let $q \in \mathbb{R}$ an arbitrary given number. If $q^n$ is irrational for all $n>1$ integer, then $q$ is irrational.
My Questions. What is a the name of this statement and what is the shortest proof for it? Thanks.

Isn't this trivial? Proof by contradiction.
Suppose $q$ is rational. Then $q^n$ is also rational for all $n>1$. Since the assumption is that they are irrational, the assumption is false and $q$ must be irrational.
Don't know if there is a name for this.
