powers of coprime numbers
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?
Solution 1:
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 $q$.
Solution 2:
$x^y$ has the same prime factors as $x$ when $y$ is an integer.