jmc

We have that $$\left|\frac{64}{5}-\frac{48}{5}i\right|=\left|\frac{16}{5}(4-3i)\right|=\frac{16}{5}|4-3i|=\frac{16}{5}(5)=16.$$We also have $|z{|}^4=|z|\cdot |z|\cdot |z|\cdot |z|=|(z)(z)(z)(z)|=|z^4|$, so $|z^4|=16$ means that $|z{|}^4=16$, which gives us $|z|=16^{\frac{1}{4}}=\boxed{2}$.