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 $$n^2-2$$.