In this paper, we investigate the dynamic behavior of a class of $p$-dimensional rational difference equation systems that extend previously studied two- and three-dimensional models. The system incorporates a power-type nonlinearity parameter $q>0$, and our analysis focuses on the boundedness of solutions and the local asymptotic stability of equilibrium points. By generalizing known results to higher dimensions, we provide a deeper understanding of the local dynamics and structural properties of such systems.