$[K:k]_i<\infty$. Show that $kK^{p^n} = kK^{p^e}$ for all $n\ge e.$

Question

I don't know where to start with this:

Let $k$ be a field of characteristic $p>0$, and let $k\subset K$ be an algebraic field extension of finite inseparable degree.

(a) Show that there exists $e\in\mathbb{N}$ such that $kK^{p^n} = kK^{p^e}$ for every $n\ge e$.

(b) Show that the inseparable degree of $k\subset K$ is $[K: kK^{p^e}]$ for $e$ as in (a).