Simplify. Assume \( z \) is greater than or equal to zero. \( 2 \sqrt{50 z^{7}} \)

To simplify the expression \( 2 \sqrt{50 z^{7}} \) with the assumption that \( z \geq 0 \), follow these steps: 1. **Factor Inside the Square Root:** Break down the expression inside the square root into perfect squares and other factors. \[ \sqrt{50 z^{7}} = \sqrt{25 \times 2 \times z^{6} \times z} \] 2. **Simplify Each Part:** - \( \sqrt{25} = 5 \) - \( \sqrt{z^{6}} = z^{3} \) (since \( \sqrt{z^{6}} = z^{6/2} = z^{3} \)) - The remaining terms under the square root are \( \sqrt{2z} \) 3. **Combine the Simplified Parts:** \[ \sqrt{50 z^{7}} = 5 z^{3} \sqrt{2z} \] 4. **Multiply by the Coefficient Outside the Square Root:** \[ 2 \times 5 z^{3} \sqrt{2z} = 10 z^{3} \sqrt{2z} \] So, the simplified form of the expression is: \[ 10 z^{3} \sqrt{2 z} \] **Final Answer:** \[ 10\, z^{3} \sqrt{\,2\, z\,} \]