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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? –  Asaf Karagila Nov 23 '12 at 12:25

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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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clearly the equivalent class of $0$ consists only of $0$. Furthermore, $(x)^2 = (-x)^2$ so without loss of generality it suffices to consider $\mathbb{R}_+$. –  Willie Wong Nov 23 '12 at 12:24
    
@WillieWong, right. I've fixed my answer... –  lhf Nov 23 '12 at 12:25

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