Simplify. Assume \( q \) is greater than or equal to zero. \( \sqrt{75 q^{3}} \)

To simplify \( \sqrt{75 q^{3}} \) with the assumption that \( q \geq 0 \), follow these steps: 1. **Factor the radicand (the expression inside the square root)**: \[ 75 q^3 = 25 \times 3 \times q^2 \times q \] 2. **Separate the square root using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)**: \[ \sqrt{75 q^3} = \sqrt{25 \times 3 \times q^2 \times q} = \sqrt{25} \times \sqrt{q^2} \times \sqrt{3q} \] 3. **Simplify each square root**: \[ \sqrt{25} = 5, \quad \sqrt{q^2} = q, \quad \text{and} \quad \sqrt{3q} \text{ remains as is since it cannot be simplified further.} \] 4. **Combine the simplified terms**: \[ 5 \times q \times \sqrt{3q} = 5q\sqrt{3q} \] **Final Simplified Form**: \[ 5q\sqrt{3q} \]