Question
Given \( x>0 \) and \( y>0 \), select the expression that is equivalent to \[ \sqrt{16 x^{7} y^{11}} \] Answer \( \begin{array}{ll}8 x^{\frac{2}{7}} y^{\frac{2}{11}} & 4 x^{\frac{2}{7}} y^{\frac{2}{11}} \\ 4 x^{\frac{7}{2}} y^{\frac{11}{2}} & 8 x^{\frac{7}{2}} y^{\frac{11}{2}}\end{array} \)
Ask by Powers Gray. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equivalent expression is \(4 x^{\frac{7}{2}} y^{\frac{11}{2}}\).
Solution
To simplify the expression \(\sqrt{16 x^{7} y^{11}}\), let's break it down step by step:
1. **Apply the square root to each factor:**
\[
\sqrt{16 x^{7} y^{11}} = \sqrt{16} \cdot \sqrt{x^{7}} \cdot \sqrt{y^{11}}
\]
2. **Simplify each square root:**
- \(\sqrt{16} = 4\)
- \(\sqrt{x^7} = x^{7/2}\) because \(\sqrt{x^n} = x^{n/2}\)
- \(\sqrt{y^{11}} = y^{11/2}\)
3. **Combine the simplified terms:**
\[
\sqrt{16 x^{7} y^{11}} = 4 \cdot x^{7/2} \cdot y^{11/2} = 4x^{7/2}y^{11/2}
\]
Among the given options, the equivalent expression is:
\[
4 x^{\frac{7}{2}} y^{\frac{11}{2}}
\]
**Answer:** \(4 x^{\frac{7}{2}} y^{\frac{11}{2}}\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify the expression \( \sqrt{16 x^{7} y^{11}} \), we can break it down into its components. 1. The square root of 16 is 4 (\( \sqrt{16} = 4 \)). 2. The square root of \( x^7 \) is \( x^{7/2} \) (using the property \( \sqrt{x^n} = x^{n/2} \)). 3. The square root of \( y^{11} \) is \( y^{11/2} \) (similarly, \( \sqrt{y^n} = y^{n/2} \)). Thus, combining all parts, we have: \[ \sqrt{16 x^{7} y^{11}} = 4 x^{7/2} y^{11/2} \] The correct equivalent expression is \( 4 x^{\frac{7}{2}} y^{\frac{11}{2}} \).
