Fix $r$ and $q$ primes with $r$ less than $q$. Find minimal $k$ such that $r\ |\ q^k-1$ and construct $F_{q^k}$ a finite field (in this case $F_{q^k}$ contains all the $r$-th roots of unity).
Define the coset of elements $(F^*_{q^k})^r = \{u^r\ |\ u \in F^*_{q^k} \}$.
Question. What is the cardinality of $(F^*_{q^k})^r$ ?
I would say it is $\frac{q^k-1}{r}$ but any hints on proving it ?