Let $k$ be a field of characteristic $p$, and let $K=k^{1/p}$. We want to show : $[A]\in\rm Br(k)$ splits over $K$, i.e. $[A\otimes_k K]=1\in\rm Br(K)$ iff $p. [A]=1\in\rm Br(k)$. Here $[A]$ denotes the class of a central simple $k$-algebra $A$ over $k$.
If $[K:k]=p$ then we clearly have $\rm exp(A)/p$ and thus $p.[A]=1$. For converse part how one should proceed?