  We establish asymptotic formulae for the number of $k$-free values of
polynmilas $F(x_1,\cdots,x_n)\in\mathbb{Z}[x_1,\cdots,x_n]$ of degree $d\geq 2$
for any $n\geq 1$, including when the variables are prime, as long as $k\geq
(3d+1)/4$. Thus we generalize a work of Browning, while we use a different
sieveing technique for the middle range of primes.
