Let k ≥ 2 be an integer and Fq be a finite field with q elements. Generalizing a recent theorem by Carmon and Entin, we find upper bounds for the size of the largest intervals free of k-th powers which are much stronger than the corresponding bounds for such intervals over the integers. We also develop polynomial versions of the classical techniques used to study gaps between k-free integers in Z. We apply these techniques to obtain analogues in Fq[x] of some classical theorems on the distribution of k-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.