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In this paper, we investigate the exact solution of an anisotropic space-time in the context of $f(Q)$ gravity, where $f\left( Q\right) $ is the arbitrary function of the non-metricity scalar $Q$. Here, we consider a specific power-law form as $f\left( Q\right) =\alpha Q^{n+1}+\beta $, where $\alpha $, $\beta $, and $n$ are free model parameters. Using power-law cosmology ($a\left( t\right) \propto t^{m}$), we analyze the physical behavior of cosmological parameters such as the energy density, pressure, EoS parameter, skewness parameter. Further, we validated the model with energy conditions and found that our $f\left( Q\right) $\ cosmological model behaves like the quintessence model in the present and $% \Lambda $CDM in the future.