We study some counting questions concerning products of positive integers u1, . . . , un which form a nonzero perfect square, or more generally, a perfect k -th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields which are multiquadratic extensions of Q and improve and generalize a result of N. Rome (2017)