A graph G = (V,E) is arbitrarily partitionable (AP) if for any sequence tau = (n_1, ..., n_p) of positive integers adding up to the order of G, there is a sequence of vertex-disjoints subsets of V whose orders are given by tau and which induce connected graphs. If, additionally, for any k, k <= p, of elements of tau we are allowed to prescribe k vertices belonging to the subsets with given size, we say that G is AP+k. We prove that the kth power of every traceable graph of order at least k is AP+(k-1) and that the kth power of every hamiltonian graph of order at least 2k is AP+(2k-1), and these results are tight.