Fraction raised to integer power

Question

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can $(p/q)^n$ ever be an integer?

Answer 1

If it is not an integer, there must be some prime which divides the denominator but not the numerator. The same will be true of all integer powers $n\ge 1$, so it cannot be an integer.

Answer 2

$(\frac{p}q)^n$ cannot be an integer for $p,q$ and $n$ integers and $n\geq 1$ and $p$ is not divisible by $q$, as in your case. For if $p$ is not divisible by $q$, then so is the case for $p^n$ and $q^n$, hence it cannot be an integer for if you want it to be an integer the denominator must be $\pm 1$, or $q$ must divide $p$. Also it trivially holds for $n=0$. Also for $n<0$ and $p$ does not divide $q$, then $(\frac{p}q)^n$ will be an integer for $p=\pm 1$ and any integer $q \neq 0 $.