We show that the eigenvalue density of the product of n identically distributed R-diagonal random matrices from a given matrix ensemble is equal to the eigenvalue density of n-th power of a single matrix from this ensemble in the limit of infinite matrix size. Using this observation one can derive the limiting eigenvalue density of the product of n independent identically distributed matrices for non-Hermitian matrix ensembles with invariant measures. We discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices.