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If $p$ and $q$ are coprime integers, does that mean the positive integral powers of $p$ and $q$ are coprime as well? I.e. $a^x = b^y$ with $a, b, x, y \in \mathbb{N}$ and $a,b$ coprime.

E.g. if $p$ and $q$ are coprime integers does that imply $p^3$ and $q^3$ are also coprime?

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    $\begingroup$ Sure, because raising to a power doesn't introduce any new prime factors. $\endgroup$
    – user4894
    Sep 20, 2014 at 19:22

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They will be because if there was a prime divisor $r$ such that $r | p^3$ and $r | q^3$, then $r|p$ and $r | q$ which will contradict the relative primality of $p$ and $q$.

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$x^y$ has the same prime factors as $x$ when $y$ is an integer.

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