  Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$.
We prove that the smallest positive integer $n$ which is a simultaneous
$p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$ n<p^{1/4 -
c_r+o(1)}\quad(p\to\infty) $$ for some positive $c_r$ satisfying $c_r\ge
e^{-(1+o(1))r} \; (r\to \infty).$
