Integral solutions of $x^\alpha+y^\alpha=z^\alpha$

Question

The problem is

Is it true that the equation $x^\alpha+y^\alpha=z^\alpha$ has no solution in integers (except $0$) where $\alpha\in \mathbb{R}\setminus\mathbb{Z}$ ?

I am for sometime with this problem but I can't solve it. If $\alpha\in \mathbb{N}$ then the problem becomes a variant of Fermat's Last Theorem but even assuming that result, can anyone give a hint (or a complete proof) as to how to proceed.

Answer 1

In fact, for any positive integers $x,y,z$ with $\max(x,y) < z$, by the Intermediate Value Theorem there is $\alpha \in (0,\infty)$ such that $x^\alpha + y^\alpha = z^\alpha$.

Answer 2

What about $9^{1/2} + 16^{1/2} = 49^{1/2}$?