We have considered the Cauchy problem for a first-order evolutionary equation with fractional powers of an operator. Such nonlocal mathematical models are used, for example, to describe anomalous diffusion processes. We want the transition to a new level in time to be solved usual problems. Computational algorithms are constructed based on some approximations of operator functions. Currently, when solving stationary problems with fractional powers of an operator, the most attention is paid to rational approximations. In the approximate solution of nonstationary problems, we come to equations with an additive representation of the problem operator. Additive-operator schemes are constructed by using different variants of splitting schemes. In the present work, the time approximations are based on approximations of the transition operator by the product of exponents. We use exponent splitting schemes of the first and second-order accuracy. The results of numerical experiments for a two-dimensional model problem with fractional powers of the elliptic operator are presented.