In the theory of latin squares and in the binary quasigroup theory the notion of a latin power set (a quasigroup power set) is known. These sets have a good property, and namely, they are orthogonal sets. Such sets were studied and methods of their construction were suggested in different articles (see, for example, [1–5]). In this article we introduce (k)-powers of a k-invertible n-ary operation (with respect to the k-multiplication of n-ary operations) and (k)-power sets of n-ary quasigroups, n ≥ 2, 1 ≤ k ≤ n, prove pairwise orthogonality of such sets and consider distinct posibilities of their construction with the help of binary groups, in particular, using n − T-quasigroups and n-ary groups.