In this paper, a new approach is successfully addressed to design the state-feedback adaptive stabilizing control law for a class of high-order nonlinear systems in triangular form and with unknown and nonidentical control coefficients, whose stabilizing control has been investigated recently under the knowledge that the lower bounds of the control coefficients are exactly known. In the present paper,without any knowledge of the lower bounds of the control coefficients, based on the adaptive technique and appropriately choosing design parameters, we give the recursive design procedure of the stabilizing control law by utilizing the approach of adding a power integrator together with tuning functions. The state-feedback adaptive control law designed not only preserves the equilibrium at the origin, but also guarantees the global asymptotic stability of the closed-loop states and the uniform boundedness of all the other closed-loop signals.