To simplify the radical expressions $\sqrt[3]{56x^5y^4}$ and $\sqrt[4]{32x^5y^8}$, apply the Product Property of Roots by factoring out the largest perfect powers that match the root index.

For the cube root $\sqrt[3]{56x^5y^4}$, identify the largest perfect cube factors: $56$ has a factor of $8$ ($2^3$), $x^5$ has $x^3$, and $y^4$ has $y^3$. Rewrite the radicand as the product of the perfect cube $8x^3y^3$ and the remaining factor $7x^2y$: $\sqrt[3]{8x^3y^3 \cdot 7x^2y}$. Split the radical and simplify the perfect cube: $\sqrt[3]{8x^3y^3} \cdot \sqrt[3]{7x^2y} = 2xy\sqrt[3]{7x^2y}$.

For the fourth root $\sqrt[4]{32x^5y^8}$, locate the largest perfect fourth power factors: $32$ has a factor of $16$ ($2^4$), $x^5$ has $x^4$, and $y^8$ is already a perfect fourth power ($(y^2)^4$). Rewrite the radicand as the product of the perfect fourth power $16x^4y^8$ and the remaining factor $2x$: $\sqrt[4]{16x^4y^8 \cdot 2x}$. Split the radical and simplify: $\sqrt[4]{16x^4y^8} \cdot \sqrt[4]{2x} = 2|x|y^2\sqrt[4]{2x}$. Absolute value signs are required around $x$ because the root index is even, ensuring the principal root is non-negative.