Question
Question Write \( \frac{\sqrt{h^{5}}}{\sqrt[6]{h^{11}}} \) as a single radical using the smallest possible root. Answer Attempt 1 out of 2 \( \sqrt[\square]{\square} \)
Ask by Romero Rodriguez. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt[3]{h^{2}}
\]
Solution
To simplify the expression \( \frac{\sqrt{h^{5}}}{\sqrt[6]{h^{11}}} \) into a single radical with the smallest possible root, follow these steps:
1. **Express Each Radical as an Exponent:**
\[
\sqrt{h^5} = h^{\frac{5}{2}} \quad \text{and} \quad \sqrt[6]{h^{11}} = h^{\frac{11}{6}}
\]
2. **Divide the Exponents:**
\[
\frac{h^{\frac{5}{2}}}{h^{\frac{11}{6}}} = h^{\frac{5}{2} - \frac{11}{6}}
\]
3. **Subtract the Exponents:**
\[
\frac{5}{2} - \frac{11}{6} = \frac{15}{6} - \frac{11}{6} = \frac{4}{6} = \frac{2}{3}
\]
So, the expression simplifies to:
\[
h^{\frac{2}{3}}
\]
4. **Convert Back to Radical Form:**
\[
h^{\frac{2}{3}} = \sqrt[3]{h^2}
\]
Thus, the simplified expression as a single radical with the smallest possible root is:
\[
\sqrt[3]{h^2}
\]
**Answer:**
\[
\sqrt[3]{h^{2}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To combine the radicals, you can express them with a common root. The first radical is \( \sqrt{h^{5}} = h^{5/2} \) and the second one is \( \sqrt[6]{h^{11}} = h^{11/6} \). Now, you subtract the exponents since you're dividing: \[ h^{5/2 - 11/6}. \] To perform the subtraction, we need a common denominator, which is 6. So, \( \frac{5}{2} = \frac{15}{6} \). Thus, \[ h^{15/6 - 11/6} = h^{4/6} = h^{2/3}. \] Now, to express \( h^{2/3} \) as a single radical, it can be written as \( \sqrt[3]{h^{2}} \). So, the final answer is \( \sqrt[3]{h^{2}} \). And there you have it!
