I will consider power substitutions in quasianalytic Carleman classes, i.e., equations of the form f(x)=g(x^k), where k>1 and f is a given function in a quasianalytic Carleman class. I will show that if g is smooth, then g belongs to a quasianalytic class completely characterized in terms of bounds on the derivatives of g. The talk is based on joint work in progress with A. Sodin.