The power domination number arose from the monitoring of electrical networks, and methods for its determination have the associated application. The zero forcing number arose in the study of maximum nullity among symmetric matrices described by a graph (and also in control of quantum systems and in graph search algorithms). There has been considerable effort devoted to the determination of the power domination number, the zero forcing number, and maximum nullity for specific families of graphs. In this paper we exploit the natural relationship between power domination and zero forcing to obtain results for the power domination number of tensor products and the zero forcing number of lexicographic products of graphs. In addition, we establish a general lower bound for the power domination number of a graph based on the maximum nullity of the matrices described by the graph. We also establish results for the zero forcing number and maximum nullity of tensor products and Cartesian products of certain graphs.