On the inverse Fermat equation (Q1199620)

The inverse Fermat equation \(x^{1/n}+ y^{1/n}=z^{1/n}\) is solved in positive integers \(x\), \(y\), \(z\), \(n\). If the \(n\)th roots are taken to be positive real numbers, it is equivalent to the following problem: if \(\alpha\), \(\beta\) are positive real numbers for which \(\alpha+\beta=1\), \(\alpha^ n\) and \(\beta^ n\) are rational, then \(\alpha\) and \(\beta\) are rational. The proof is very short called a ``one line proof''. If the \(n\)th roots rather than positive real ones are allowed in the equation, the author proves that up to a suitable equivalence relation there are exactly four nontrivial solutions.