The number of states in a deterministic finite automaton (DFA) recognizing the language L^k, where L is regular language recognized by an n-state DFA, and k ≥ 2 is a constant, is shown to be at most n 2^{(k - 1) n} and at least (n - k) 2^{(k - 1) (n - k)} in the worst case, for every n > k and for every alphabet of at least six letters. Thus, the state complexity of L^k is Θ (n 2^{(k - 1) n}). In the case k = 3 the corresponding state complexity function for L³ is determined as frac(6 n - 3, 8) 4ⁿ - (n - 1) 2ⁿ - n with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of L^k is demonstrated to be n k. This bound is shown to be tight over a two-letter alphabet.