Simplify. Assume \( a \) is greater than or equal to zero. \( \sqrt{75 a^{10}} \)

To simplify \( \sqrt{75 a^{10}} \), we first break it down into its components. We have: 1. The number \( 75 \) can be expressed as \( 25 \times 3 \), so we rewrite the expression as: \[ \sqrt{75 a^{10}} = \sqrt{25 \times 3 \times a^{10}}. \] 2. Next, we use the property of square roots \( \sqrt{xy} = \sqrt{x} \cdot \sqrt{y} \). Thus, we can split this into: \[ \sqrt{75 a^{10}} = \sqrt{25} \cdot \sqrt{3} \cdot \sqrt{a^{10}}. \] 3. Simplifying these square roots gives: \[ \sqrt{25} = 5 \quad \text{and} \quad \sqrt{a^{10}} = a^{5} \quad \text{(since \( a \) is non-negative)}. \] 4. Combining all parts, we get: \[ 5 \cdot \sqrt{3} \cdot a^{5} = 5a^{5} \sqrt{3}. \] So, the simplified form of \( \sqrt{75 a^{10}} \) is: \[ \boxed{5a^{5} \sqrt{3}}. \]