The topic of this paper is the computation of the matrix exponential in the context of burnup equations. The established matrix exponential methods are introduced briefly. The eigenvalues of the burnup matrix are important in choosing the matrix exponential method, and their characterization is considered. Based on the characteristics of the burnup matrix, the Chebyshev rational approximation method (CRAM) and its interpretation as a numeric contour integral are discussed in detail. The introduced matrix exponential methods are applied to two test cases representing an infinite pressurized water reactor pin-cell lattice, and the numerical results are presented. The results suggest that CRAM is capable of providing a robust and accurate solution to the burnup equations with a very short computation time.