QuestionAugust 25, 2025

Question Express in simplest radical form. 5xsqrt [3](162x^5y^6)

Question Express in simplest radical form. 5xsqrt [3](162x^5y^6)
Question Express in simplest radical form. 5xsqrt [3](162x^5y^6)

Solution
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Answer

15x^2y^2 \sqrt[3]{6x^2} Explanation 1. Simplify the Radicand Break down 162x^5y^6 into prime factors: 162 = 2 \times 3^4, so 162x^5y^6 = 2 \times 3^4 \times x^5 \times y^6. 2. Apply Cube Root to Each Factor Use the property \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}. Calculate cube roots: \sqrt[3]{2}, \sqrt[3]{3^4} = 3\sqrt[3]{3}, \sqrt[3]{x^5} = x^{5/3}, \sqrt[3]{y^6} = y^2. 3. Combine Simplified Terms Combine terms: 3xy^2 \cdot \sqrt[3]{2 \cdot 3 \cdot x^2}. 4. Multiply by the Coefficient Multiply by 5x: 5x \cdot 3xy^2 \cdot \sqrt[3]{6x^2} = 15x^2y^2 \cdot \sqrt[3]{6x^2}.

Explanation

1. Simplify the Radicand<br /> Break down $162x^5y^6$ into prime factors: $162 = 2 \times 3^4$, so $162x^5y^6 = 2 \times 3^4 \times x^5 \times y^6$.<br /><br />2. Apply Cube Root to Each Factor<br /> Use the property $\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$.<br /> Calculate cube roots: $\sqrt[3]{2}$, $\sqrt[3]{3^4} = 3\sqrt[3]{3}$, $\sqrt[3]{x^5} = x^{5/3}$, $\sqrt[3]{y^6} = y^2$.<br /><br />3. Combine Simplified Terms<br /> Combine terms: $3xy^2 \cdot \sqrt[3]{2 \cdot 3 \cdot x^2}$.<br /><br />4. Multiply by the Coefficient<br /> Multiply by $5x$: $5x \cdot 3xy^2 \cdot \sqrt[3]{6x^2} = 15x^2y^2 \cdot \sqrt[3]{6x^2}$.
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