Is x^{\sqrt{5}}=y for Rational Numbers?

[Gerenuk]

I've programmed an algorithm to numerically compute the logarithm of numbers in phinary base easily. I could avoid float multiplications if I can find a pair of rational numbers x and y such that
$$x^{\sqrt{5}}=y$$
Is it possible?
Probably not, but I cannot prove it :(

 

[gel]

No, it isn't possible except for the obvious x=y=1. There's a http://en.wikipedia.org/wiki/Gelfond–Schneider_theorem" which says that it's impossible.