Unseparable element becomes separable

Question

Let $L/K$ be an extension of fields. Denote by $p$ the characteristic of $K$. One assumes that $p$ is a a prime. Let $a\in L$ not separable over $K$. Does it exist an integer $s$ such $a^{p^s}$ is separable over $K$ and $[K(a^{p^s}):K]<p$?