This paper presents an algorithm for least squares approximation of higher-order tensors by rank-1 tensors. The higher-order power method, related to the method of iterating powers of a matrix, is an iteration using weighted Tucker products. This paper gives a symmetric version of this method for supersymmetric tensors and proves it converges when an associated function is convex or concave. It also gives evidence that this method is significantly faster for supersymmetric tensors.