 Coprime-degree descent for powers

Let $L/K$ have degree $d$ with $\gcd (d,n)=1$. If $a=x^n\in K$ for some $x\in L$, thenChoosing $r,s\in \mathbb{Z}$ with $rd+sn=1$ givesso $a$ was already an $n$th power in $K$.