The role of Darrieus-Landau instability in forming the fractal structure of the freely propagating expanding flame front is studied numerically by solving the two-dimensional Navier-Stokes equations with one step irreversible Arrhenius reaction. Numerical simulation of the radially expanding flames with Le=1 demonstrates that the temporal dependence of mean radius after a certain critical time instant is corresponding to a power law, in line with previous experimental, numerical and theoretical studies. Influence of gas expansion ratio on power-law exponent and flame surface fractal dimension is the focus of this research. It is shown that the expansion coefficient has an effect on fractal structure of the outwardly propagating flame, confirming the assumption that fractal dimension is not a universal parameter.