Define the relation $x \sim y$ where $x$ and $y$ are real numbers to hold if and only if there exist natural numbers $n$ and $m$ such that $x^n = y^m$.
It is easy to see that $\sim$ is an equivalence relation. My question is if this equivalence relation naturally shows up in some setting and perhaps has been named (or perhaps the equivalence classes of it have been named)?
I don't know that this relation shows up naturally, but one natural name for it is the squares have commensurable logarithms (in any and all bases).