To simplify higher-order numerical roots, use the Product Property of Roots to extract the largest perfect power factor corresponding to the index of each radical. For the square root $\sqrt{432}$, the largest perfect square factor is $144$, which simplifies to $\sqrt{144 \cdot 3} = 12\sqrt{3}$. For the cube root $\sqrt[3]{625}$, the greatest perfect cube factor is $125$ (since $5^3 = 125$), resulting in $\sqrt[3]{125 \cdot 5} = 5\sqrt[3]{5}$. For the fourth root $\sqrt[4]{729}$, the largest perfect fourth power factor is $81$ (since $3^4 = 81$), yielding $\sqrt[4]{81 \cdot 9} = 3\sqrt[4]{9}$.