Question
Simplify. Assume \( z \) is greater than or equal to zero. \( 6 \sqrt{20 z^{7}} \)
Ask by Lindsey Bernard. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(12 z^{3} \sqrt{5z}\)
Solution
To simplify \( 6 \sqrt{20 z^{7}} \) with \( z \geq 0 \), follow these steps:
1. **Factor Inside the Square Root:**
\[
\sqrt{20 z^{7}} = \sqrt{4 \times 5 \times z^{6} \times z}
\]
2. **Simplify Each Part:**
- \( \sqrt{4} = 2 \)
- \( \sqrt{z^{6}} = z^{3} \) (since \( z \geq 0 \))
- \( \sqrt{5z} \) remains as it is.
3. **Combine the Simplified Parts:**
\[
\sqrt{20 z^{7}} = 2 \times z^{3} \times \sqrt{5z} = 2 z^{3} \sqrt{5z}
\]
4. **Multiply by the Coefficient Outside the Square Root:**
\[
6 \times 2 z^{3} \sqrt{5z} = 12 z^{3} \sqrt{5z}
\]
**Final Simplified Form:**
\[
12 z^{3} \sqrt{5z}
\]
Answer: \(12\, z^{3} \sqrt{\,5\, z\,}\)
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Bonus Knowledge
To simplify \( 6 \sqrt{20 z^{7}} \), we start by breaking down the expression inside the square root: \[ 20 z^{7} = 4 \cdot 5 \cdot z^{6} \cdot z = 4 z^{6} \cdot 5 z \] Now, let's take the square root: \[ \sqrt{20 z^{7}} = \sqrt{4 z^{6} \cdot 5z} = \sqrt{4} \cdot \sqrt{z^{6}} \cdot \sqrt{5z} \] Calculating each part gives us: \[ \sqrt{4} = 2, \quad \sqrt{z^{6}} = z^{3}, \quad \text{and } \sqrt{5z} = \sqrt{5} \sqrt{z} \] So we have: \[ \sqrt{20 z^{7}} = 2 z^{3} \sqrt{5z} \] Now, we multiply this by 6: \[ 6 \sqrt{20 z^{7}} = 6 (2 z^{3} \sqrt{5z}) = 12 z^{3} \sqrt{5z} \] Thus, the simplified expression is: \[ \boxed{12 z^{3} \sqrt{5z}} \]
