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 ?

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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 ?

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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.