All fractions which exponentiated by another fraction gives yet another fraction

Question

Consider $\left(\dfrac{a}{b}\right)^{\dfrac{c}{d}}=\dfrac{e}{f}$, where $a, b, c, d, e, f \in \mathbb{Z}$ (the fractions need not be irreducible). Which are all $a, b, c, d$? Note: I'm not interested in $e, f$.

Answer 1

This can give a rational result only if $a / b$ is an exact $d$-th power of a fraction, i.e., if the reduced fraction can be written:

$\begin{align*} \frac{a}{b} &= \frac{u^d}{v^d} \end{align*}$

for integers $u, v$; the result is then just:

$\begin{align*} \left(\frac{a}{b}\right)^{\frac{c}{d}} &= \frac{u^c}{v^c} \end{align*}$