In a recent paper, Soundararajan has proved the quantum unique ergodicity conjecture by getting a suitable estimate for the second order moment of the so-called ‘Hecke multiplicative’ functions. In the process of proving this he has developed many beautiful ideas. In this paper we generalize his arguments to a general kth power and provide an analogous estimate for the kth power moment of the Hecke multiplicative functions. This may be of general interest.