Question
Simplify. Assume \( y \) is greater than or equal to zero. \( 9 \sqrt{8 y} \)
Ask by Valdez Schultz. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 18 \sqrt{2y} \)
Solution
To simplify the expression \( 9 \sqrt{8y} \) with the condition that \( y \geq 0 \), follow these steps:
1. **Factor the Radicand:**
Start by factoring the number inside the square root to simplify the expression.
\[
\sqrt{8y} = \sqrt{4 \times 2y}
\]
2. **Simplify the Square Root:**
Use the property of square roots that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
\[
\sqrt{4 \times 2y} = \sqrt{4} \times \sqrt{2y} = 2 \sqrt{2y}
\]
3. **Multiply by the Coefficient:**
Now, multiply the simplified square root by the coefficient outside the square root (which is 9).
\[
9 \times 2 \sqrt{2y} = 18 \sqrt{2y}
\]
**Final Simplified Form:**
\[
18 \sqrt{2y}
\]
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Simplify this solution Extra Insights
To simplify \( 9 \sqrt{8y} \), we can break it down into its components. First, note that \( \sqrt{8} \) can be simplified as \( \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \). Therefore, we have: \[ \sqrt{8y} = \sqrt{8} \cdot \sqrt{y} = 2\sqrt{2} \cdot \sqrt{y} \] Now substituting this back into the expression gives: \[ 9 \sqrt{8y} = 9(2\sqrt{2} \cdot \sqrt{y}) = 18\sqrt{2y} \] So, the simplified form is: \[ \boxed{18\sqrt{2y}} \]
