jmc

Since $z^3=100+75i$, we must have $|z^3|=|100+75i|=|25(4+3i)|=25|4+3i|=25(5)=125$. We also have $|z{|}^3=|z|\cdot |z|\cdot |z|=|(z)(z)(z)|=|z^3|$, so $|z^3|=125$ means that $|z{|}^3=125$, which gives us $|z|=\sqrt[3]{125}=\boxed{5}$.