We examine the deterministic and nondeterministic state complexity of square, power, positive closure, and complementation on star-free languages. For the state complexity of square, we get a non-trivial upper bound 

$$(n-1)2^n - 2(n-2)$$ and a lower bound of order  $${\Theta }(2^n)$$ . For the state complexity of the k-th power in the unary case, we get the tight upper bound  $$k(n-1)+1$$ . Next, we show that the upper bound kn on the nondeterministic state complexity of the k-th power is met by a binary star-free language, while in the unary case, we have a lower bound  $$k(n-1)+1$$ . For the positive closure, we show that the deterministic upper bound  $$2^{n-1}+2^{n-2}-1$$ , as well as the nondeterministic upper bound n, can be met by star-free languages. We also show that in the unary case, the state complexity of positive closure is  $$n^2-7n+13$$ , and the nondeterministic state complexity of complementation is between  $$(n-1)^2+1$$ and  $$n^2-2$$ .