On the equation xk=z1k1z2k2⋯znkn in a free semigroup

Word equations of the form xk=z1k1z2k2⋯znkn are considered in this paper. In particular, we investigate the case where x is of different length than zi, for any i, and k and ki are at least 3, for all 1⩽i⩽n, and n⩽k. We prove that for those equations all solutions are of rank 1, that is, x and zi are powers of the same word for all 1⩽i⩽n. It is also shown that this result implies a well-known result by Appel and Djorup about the more special case where ki=kj for all 1⩽i