1Cademy - Simplifying $$\sqrt[3]{56x^5y^4}$$ and $$\sqrt[4]{32x^5y^8}$$

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Simplifying $\sqrt[3]{40x^4y^5}$ and $\sqrt[4]{48x^4y^7}$

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Simplifying $\sqrt[3]{56x^5y^4}$ and $\sqrt[4]{32x^5y^8}$

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.

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Updated 2026-05-26

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OpenStax Intermediate Algebra

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