Let p 1 , . . . , p s be distinct primes, and S the set of integers not divisible by primes different from p 1 , . . . , p s . We give effectively computable upper bounds for n in the equations (1) f (x) = wy n , (5) F (x, z) = wy n and (9) ax n − by n = c, where f ∈ Z[X] is a monic polynomial, F ∈ Z[X, Z] a monic binary form, the discriminants D(f ), D(F ) are contained in S, and x, y, z, w, a, b, c, n are unknown non-zero integers with z, w, a, b, c ∈ S, y / ∈ S and n ≥ 3. It is a novelty in our paper that the upper bounds depend only on the product p 1 • • • p s and, in case of (1) and ( 5), on deg f and deg F , respectively. The bounds are given explicitly in terms of p 1 • • • p s . Our results are established in more general forms, over an arbitrary algebraic number field. Equation ( 5) is reduced to an equation of type (9) over an appropriate extension of the ground field. The proofs involve among other things the best known estimates for linear forms in logarithms of algebraic numbers.