Question
Multiple Choice 1 point Which expression is equivalent to \( \sqrt{36 x^{9} y^{25}} \), where \( x>0 \) and \( y>0 \)
Ask by Simmons Ruiz. in the United States
Jan 08,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression simplifies to \(6x^{4}y^{12}\sqrt{xy}\).
Solution
To simplify the expression \( \sqrt{36 x^{9} y^{25}} \) where \( x > 0 \) and \( y > 0 \), follow these steps:
1. **Separate the Square Root:**
\[
\sqrt{36 x^{9} y^{25}} = \sqrt{36} \times \sqrt{x^{9}} \times \sqrt{y^{25}}
\]
2. **Simplify Each Component:**
- \( \sqrt{36} = 6 \)
- \( \sqrt{x^{9}} = x^{4} \sqrt{x} \) (since \( x^{9} = x^{8} \times x = (x^{4})^2 \times x \))
- \( \sqrt{y^{25}} = y^{12} \sqrt{y} \) (since \( y^{25} = y^{24} \times y = (y^{12})^2 \times y \))
3. **Combine the Simplified Parts:**
\[
6 \times x^{4} \times y^{12} \times \sqrt{x y} = 6 x^{4} y^{12} \sqrt{xy}
\]
So, the expression simplifies to:
**Answer:**
\(6\,x^{4}\,y^{12}\) multiplied by the square root of \(x y\), that is
\(6x^{4}y^{12}\sqrt{xy}\).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To simplify the expression \( \sqrt{36 x^{9} y^{25}} \), we can break it down into parts. The square root of \( 36 \) is \( 6 \). For \( x^9 \), since we know \( \sqrt{x^2} = x \) and \( \sqrt{x^4} = x^2 \), we can rewrite \( x^9 \) as \( x^{4.5} \) or \( x^4 \cdot x^{0.5} \). This gives us \( \sqrt{x^9} = x^{4.5} = x^4 \sqrt{x} \). For \( y^{25} \), we have \( \sqrt{y^{25}} = y^{12.5} = y^{12} \sqrt{y} \). Putting it all together, the expression becomes \( 6x^4y^{12}\sqrt{x}\sqrt{y} \). So, the simplified expression is \( 6x^4y^{12}\sqrt{xy} \).
