Our goal is to study fractional powers of a cascade system of partial differential equations. We explicitly calculate the fractional powers of linear operators associated to this type of system and we discuss local solvability of the fractional equation with subcritical nonlinearity. As an example, a cascade system of Schrödinger equation is analyzed. A connection between the fractional system and the original system is established and we prove the convergence of the linear semigroups obtained by the fractional power operator to the original linear semigroup, as the power α approaches 1.