jmc

First of all, we know that $|ab|=|a|\cdot |b|$, so $$\left|{\left(3+\sqrt{7}i\right)}^3\right|={\left|3+\sqrt{7}i\right|}^3$$We also find that $$\left|3+\sqrt{7}i\right|=\sqrt{{\left(3\right)}^2+{\left(\sqrt{7}\right)}^2}=\sqrt{16}=4$$Therefore, our answer is $4^3=\boxed{64}$.