TITLE: Enquiry on a Diophantine problem
QUESTION [9 upvotes]: Let $x,y, z$ be relatively prime integers with $xyz \neq 0$. Suppose that
$$x^{m/n} + y^{m/n} = z^{m/n}$$ 
where $m,n$ are relatively prime integers with $mn \neq 0$.
Does it necessarily follow that $x,y,z$ are perfect $n$-th powers ?

REPLY [13 votes]: Yes, and much more is true. There are no non-trivial dependencies between radicals. See
Besicovitch A. S., On the linear independence of fractional powers of integers
// J. London Math. Soc. 15 (1940), 3–6.