Let there be two natural numbers $z$ and $w$ each raised to the same positive whole number power $n$ greater than $1$. Then let $z^n=a r$ and $w^n=a s$, with $a$ being a common natural number factor.
On calculating the $n$'th root of $\frac{z^n}{w^n}$ we have
$$\sqrt[^n]{\frac{z^n}{w^n}}=\sqrt[^n]{\frac{a r}{a s}}=\sqrt[^n]{\frac{ r}{s}}=\frac{ \sqrt[^n]{r} }{\sqrt[^n]{s}}=\frac{z}{w}(= \mathbb{Q})$$
Since $r$ and $s$ are whole numbers; is $a=t^n$, $r=u^n$ and $s=v^n$, where $t$, $u$ and $v$ are all whole numbers as well, the only solution?
If so I am not sure how to prove by contradiction that no other solutions exist, for example where a common irrational factor to $\sqrt[^n]{r}$ and $\sqrt[^n]{s}$ cancels out, leaving the rational number answer?
Do I have to add the constraint that when $a$ is taken as the common factor, that $r$ and $s$ must be left relatively prime?