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This article gives a numerical study of tangent hyperbolic nanofluid flow across an exponentially stretched sheet by considering two heat transfer situations: prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux (PEHF). Tangent hyperbolic nanofluid is being used extensively in the cooling of electronic components, which generate a lot of heat during operation. The improved thermal conductivity of tangent hyperbolic nanofluids can help to dissipate the heat more effectively, reducing the risk of overheating and component failure. A mathematical model of the problem is based on conservation laws of momentum, mass, and energy. The governing system of nonlinear PDEs is transformed into a system of nonlinear ODEs using appropriate similarity transformations. MATLAB's bvp4c tool was utilised to address the converted system of modelled equations. The drag coefficient, heat transfer, and mass transfer rate values are tabulated. It is observed that both the magnetic parameter (M) and Weissenberg number (We) causes to reduce the boundary layer. It is also noticed that increasing the Brownian motion parameter (Nb) and the thermophoresis parameter (Nt) leads the temperature θ(η) and concentration profile g(η) to increase for both cases of heat transfer i.e., PEST and PEHF.