In this letter, we investigate the cosmic expansion scenario of universe in framework $f(R,L_m)$ gravity theory. We consider a non-linear model, specifically, $f(R,L_m)=\frac{R}{2}+L_m^n + \beta$, where $n$ and $\beta$ are free model parameters. Then derive motion equations for flat FLRW obtain exact solution corresponding field equations. estimate best fit ranges parameters by using updated $H...