How to prove an irrational exponential number with its exponent divisors also leading to irrational exponential numbers

Question

If $x^n$ is irrational for some $n ∈ N$, how do I prove that $x^d$ is irrational for any divisor $d$ of $n$?

I'm thinking of using a proof of contradiction for something like this. I know how to prove just a normal base number irrational, by equaling it to $a/b$ doing proof by contradiction, but having an exponent factored into it makes it a but confusing for me.

Answer 1

Suppose $x^d$ is rational. Then $(x^d)^c=x^{dc}=x^n$ (where $dc=n$, since $d\mid n$) must also be rational, which is a contradiction. Thus $x^d$ is irrational.