This paper is focused on the study for the effects of constant heat flux and thermal buoyancy on the steady two-dimensional flow and heat transfer of a non-Newtonian power-law fluid over a non-linearly stretching vertical surface in the presence of thermal radiation. Highly nonlinear momentum and thermal boundary layer equations which governing the flow and heat transfer are reduced to a set of nonlinear ordinary differential equations by appropriate transformation. The resulting ODEs are successfully solved numerically with the help of fourth order Runge–Kutta method coupled with the shooting technique. The effects of various parameters like the buoyancy (mixed convection) parameter, the radiation parameter, power-law index parameter and the local Prandtl number on the flow and temperature profiles as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. Favorable comparisons of numerical results with previously published work on various special cases of the problem are obtained.