A computational study on the stagnation-point flow of an electrically conducting nanofluid over a nonlinear stretching/shrinking surface with first-order slip phenomenon is carried out. Because, nanofluids are considered as one of the closest kinds to the practical application of fluid flows owing to its comprehensive properties such as the Brownian motion and thermophoresis. The simulations were performed to understand the heat and mass transfer traits in the presence of non-linear radiation. Moreover, the Rosseland approximation model is incorporated to investigate the mechanisms of radiative heat transfer performance of nanofluids. For all cases, it has been seen that the governing conservation equations of current model possesses a similarity solution. The transformed system of nonlinear ordinary differential equations, is then integrated numerically with a boundary value problem solver, bvp4c in Matlab software. The captured numerical results have been displayed graphically and some interesting features like multiple (upper and lower) solutions are found. The critical values corresponding to the suction parameter integral w and the shrinking parameter lambda are computed. The rising thermo-physical dimensionless parameters overseeing the flow are power-law parameter, Hartmann number, slip parameter, Eckert number, temperature ratio parameter, radiation parameter, Brownian motion and thermophoresis parameters and Lewis number. It is found that, the slip parameter has a reducing impact on the skin friction coefficient for both upper and lower branch solutions. The major outcome of the present study is that the temperature ratio parameter boosts the temperature profiles for both solutions. While the temperature profiles show a decreasing behavior for higher non-linear radiation parameter. Validation of numerical scheme is accomplished by means of benchmarking with some already reported studies, and a great correlation is illustrated. (C)2017 Elsevier Ltd. All rights reserved.