Simplify. Assume \( d \) is greater than or equal to zero. \[ \sqrt{45 d^{10}} \]

To simplify the expression \(\sqrt{45 d^{10}}\) with the assumption that \(d \geq 0\), follow these steps: 1. **Separate the Square Root:** \[ \sqrt{45 d^{10}} = \sqrt{45} \cdot \sqrt{d^{10}} \] 2. **Simplify \(\sqrt{d^{10}}\):** Since \(d \geq 0\), \[ \sqrt{d^{10}} = d^{5} \] 3. **Simplify \(\sqrt{45}\):** Break down 45 into its prime factors: \[ 45 = 9 \times 5 = 3^2 \times 5 \] Therefore, \[ \sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5} \] 4. **Combine the Simplified Parts:** \[ \sqrt{45 d^{10}} = 3\sqrt{5} \cdot d^5 = 3d^5\sqrt{5} \] **Final Simplified Form:** \[ 3d^{5} \sqrt{5} \] Answer: \(3\, d^{5} \sqrt{5}\)