Let $F$ be a field of characteristic $0$, and let $p$ be a prime number. Choose an element $r\in F$ and suppose that $r$ has no $p$'th root in $F$. Let $F':=F(\sqrt[p]{r})$, this is an extension of $F$ of degree $p$. Set $\alpha:=\sqrt[p]{r}$ for convenience.
My question is this: Given an element $\beta\in F'$ such that $\beta^p\in F$, is it true that $\beta=c\cdot\alpha^k$ for some $c\in F$, $k\in\mathbb{N}$? In other words, it it true that the only elements of $F$ that have $p$'th roots in $F(\sqrt[p]{r})$ are the "obvious" elements $c^p\cdot r^k$?
This is true if $p=2$ or $3$, which I discerned through direct computation, which would become impractical for higher values of $p$. If anyone has any ideas for how to approach the general case, it would be very helpful.