QuestionJuly 20, 2025

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. sqrt [4]((13x)/(243y^16))

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. sqrt [4]((13x)/(243y^16))
Use the quotient rule to simplify. Assume that all variables represent positive real numbers. sqrt [4]((13x)/(243y^16))

Solution
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Answer

\frac{13^{\frac{1}{4}} \cdot x^{\frac{1}{4}}}{3 \cdot y^4} Explanation 1. Apply the fourth root Rewrite the expression as \left(\frac{13x}{243y^{16}}\right)^{\frac{1}{4}}. 2. Distribute the exponent Apply the exponent to both numerator and denominator: \frac{(13x)^{\frac{1}{4}}}{(243y^{16})^{\frac{1}{4}}}. 3. Simplify each part Simplify the numerator: (13x)^{\frac{1}{4}} = 13^{\frac{1}{4}} \cdot x^{\frac{1}{4}}. Simplify the denominator: 243^{\frac{1}{4}} \cdot (y^{16})^{\frac{1}{4}} = 3 \cdot y^4. 4. Combine results The simplified form is \frac{13^{\frac{1}{4}} \cdot x^{\frac{1}{4}}}{3 \cdot y^4}.

Explanation

1. Apply the fourth root<br /> Rewrite the expression as $\left(\frac{13x}{243y^{16}}\right)^{\frac{1}{4}}$.<br />2. Distribute the exponent<br /> Apply the exponent to both numerator and denominator: $\frac{(13x)^{\frac{1}{4}}}{(243y^{16})^{\frac{1}{4}}}$.<br />3. Simplify each part<br /> Simplify the numerator: $(13x)^{\frac{1}{4}} = 13^{\frac{1}{4}} \cdot x^{\frac{1}{4}}$.<br /> Simplify the denominator: $243^{\frac{1}{4}} \cdot (y^{16})^{\frac{1}{4}} = 3 \cdot y^4$.<br />4. Combine results<br /> The simplified form is $\frac{13^{\frac{1}{4}} \cdot x^{\frac{1}{4}}}{3 \cdot y^4}$.
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