On the difference between perfect powers and integral $S$-units

Let $q_1,\dots ,q_t$ be distinct prime numbers. Let $a_1,\dots ,a_t$ be nonnegative integers. We establish effective lower bounds for $\mathrm{\mid }{z}^d-q_1^{a_1}\dots q_t^{a_t}\mathrm{\mid }$ and for its greatest prime factor, which tend to infinity with $z^d$, where $z$ is a positive integer coprime with $q_1\dots q_t$ and $d\ge 2$ is an integer.