jmc

To start removing the radicals, we square both sides of the equation. This gives $$Q^3={\left(\sqrt{Q^3}\right)}^2={\left(16\sqrt[8]{16}\right)}^2=256\cdot \sqrt[4]{16}=256\cdot 2=512.$$Thus, $Q=\sqrt[3]{512}=\sqrt[3]{2^9}=\boxed{8}.$