$a,b,c \in \mathbb{Q^{+}}$, $abc=1$. There are $x,y,z\in \mathbb{N}\smallsetminus \{0\}$, such that $a^{x}+b^{y}+c^{z}\in \mathbb{Z}$. The task is to prove that $a,b,c$ can be expressed as $\frac{m^{k}}{n}$ for coprime $m,n$, while $k\gt 1$.
I expanded the equations and tried to deduce some useful relationships but without much success. The condition is obviously false for whole numbers which got me thinking about the properies of rationals expressable in such a manner. Do you know of any neat properties that might be relevant in this problem? Are they a well known mathematical structure in general?
Thanks for any help with the proof.