integration - Gaussian Integral

Consider the following Gaussian Integral $$I = \int_{-\infty}^{\infty} e^{-x^2} \ dx$$

The usual trick to calculate this is to consider $$I^2 = \left(\int_{-\infty}^{\infty} e^{-x^2} \ dx \right) \left(\int_{-\infty}^{\infty} e^{-y^{2}} \ dy \right)$$

and convert to polar coordinates. We get $\sqrt{\pi}$ as the answer.

Is it possible to get the same answer by considering $I^{3}, I^{4}, \dots, I^{n}$?