Let $K$ be a field of characteristic $p$ and consider a polynomial $p(x) = x^n - a \in K[x]$, with $a \neq 0$ and $p \nmid n$. By the derivative test this is certainly separable.
Even if $p$ does not split in $K$, is it true that the prime factors of $p(x)$ in $K[x]$ (a UFD) are all distinct?
My field theory is fuzzy, but it seems that if any prime factor occurred multiple times, then in a splitting field the roots of that prime factor would all occur multiple times, violating separability. Is this valid reasoning?
Let $F$ be the splitting field of $p(x)$ over $K$. If $p(x)$ had repeated prime factors in $K[x]$, then $p(x)$ would have repeated roots in $F[x]$ by uniqueness of factorization. This would contradict the fact that $p(x)$ is separable as you have already shown.