Simplify. Assume \( z \) is greater than or equal to zero. \( 6 \sqrt{20 z^{7}} \)

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: \(12\, z^{3} \sqrt{\,5\, z\,}\)