  Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime
number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the
polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods
to construct explicitly elements of high order in the field $F_q[x]/\langle
x^m-a\rangle $. Namely, we find elements with multiplicative order of at least
$5^{\sqrt[3]{m/2}}$, which is better than previously obtained bound for such
family of extension fields.
