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This paper addresses the problem of global state feedback stabilization for a class of genuinely nonlinear systems with a time-varying power. By revamping the so-called adding a power integrator technique and the homogeneous domination approach, a new design method called interval homogeneous domination approach is proposed to delicately design a state feedback control law that renders the nonlinear systems globally asymptotically stable. The novelty of the proposed scheme owes to the systematic fashion that provides a distinct perspective to solve the stabilization problem for the nonlinear systems with a time-varying power.