Fermat Last Theorem for non Integer Exponents

Suppose $z> \max(x,y)$ then $x^0+y^0 = 2 > z^0$ but there exists some $N$ such that $x^N+y^N<z^N$. Therefore there exists some $n\in[0,N]$ satisfying $x^n+y^n=z^n$.

Take $x=4, y=9$ and $n = 0.5$. You can solve to get $z = 25$. So this works!

$1782^n + 1841^n = 1922^n$ with $n \approx 11.999999995097161$