State complexity of power

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 n2^(^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 @Q(n2^(^k^-^1^)^n). In the case k=3 the corresponding state complexity function for L^3 is determined as 6n-384^n-(n-1)2^n-n with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of L^k is demonstrated to be nk. This bound is shown to be tight over a two-letter alphabet.