Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Question

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition:

• $x=n^r$ is an irrational number
• $y=m^r$ is an irrational number

Question: is $\frac{x}{y}$ an irrational (neither rational or integer)?

Answer 1

Hint: Yet another use of the rational root theorem, $x/y = (n/m)^r$ solves $z^q - (n^p/m^p) =0$, now with leading coefficient $\neq 1$.