Factor $147=7^{2}\times 3$. Rewrite the square root of the product $\sqrt{7^{2}\times 3}$ as the product of square roots $\sqrt{7^{2}}\sqrt{3}$. Take the square root of $7^{2}$.
$$7\sqrt{3}+\sqrt{108}$$
Factor $108=6^{2}\times 3$. Rewrite the square root of the product $\sqrt{6^{2}\times 3}$ as the product of square roots $\sqrt{6^{2}}\sqrt{3}$. Take the square root of $6^{2}$.
$$7\sqrt{3}+6\sqrt{3}$$
Combine $7\sqrt{3}$ and $6\sqrt{3}$ to get $13\sqrt{3}$.
