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Simplify \( \frac{\sqrt[11]{x^{9}}}{\sqrt[4]{x^{3}}} \) \( X^{[?]} \)

Ask by Wilson George. in the United States
Jan 21,2025

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\( x^{\frac{3}{44}} \)

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The Deep Dive

To simplify \( \frac{\sqrt[11]{x^{9}}}{\sqrt[4]{x^{3}}} \), we start by rewriting the roots as exponents: The numerator becomes \( x^{9/11} \) and the denominator becomes \( x^{3/4} \). Thus, we have \( \frac{x^{9/11}}{x^{3/4}} \). When dividing exponents with the same base, we subtract the exponents: \[ x^{9/11 - 3/4} \] To subtract \( \frac{3}{4} \) from \( \frac{9}{11} \), we need a common denominator, which is 44. Converting, we have: \[ \frac{9}{11} = \frac{36}{44} \quad \text{and} \quad \frac{3}{4} = \frac{33}{44} \] Subtracting these gives: \[ \frac{36}{44} - \frac{33}{44} = \frac{3}{44} \] So, the expression simplifies to \( x^{3/44} \). Final answer: \( x^{\frac{3}{44}} \)

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