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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)?

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    I assume that in this context $0$ is not a natural number?2012-11-23

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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).

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    @WillieWong, right. I've fixed my answer...2012-11-23