Exponential sums with consecutive modular roots of an integer

J. Bourgain and the author have recently estimated exponential sums with consecutive modular roots θ^1/n (mod p), where θ is of multiplicative order t ≥ p^ε modulo a prime p (for some fixed ε > 0) and n runs through the integers in the interval [M + 1, M + N] with gcd(n, t) = 1. However, the saving in that bound against the trivial estimate has not been made explicit. It is shown here that for t ≥ p ^1/2+ε one can obtain a fully explicit bound for such exponential sums.