A generalized Cauchy type functional equation

Question

Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.

Then is it true that $f(x+y)=f(x)+f(y),\forall x,y\in S$ ?