Which of the following expressions is equivalent to \( \sqrt{75 x} \) ? ( \( 5 x)^{\frac{1}{2}} \) ( \( (5 x)^{\frac{75}{2}} \) ( \( 5(3 x)^{\frac{1}{2}} \) D \( 5(25 x)^{\frac{1}{3}} \)

To determine which expression is equivalent to \( \sqrt{75x} \), let's simplify \( \sqrt{75x} \): 1. **Factor the expression inside the square root:** \[ 75x = 25 \times 3x \] 2. **Apply the property of square roots that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \):** \[ \sqrt{75x} = \sqrt{25 \times 3x} = \sqrt{25} \times \sqrt{3x} \] 3. **Simplify the square root of 25:** \[ \sqrt{25} = 5 \] So, \[ \sqrt{75x} = 5 \times \sqrt{3x} = 5(3x)^{\frac{1}{2}} \] Therefore, the expression \( \sqrt{75x} \) is equivalent to: **\( 5(3x)^{\frac{1}{2}} \)** **Answer:** \( 5(3x)^{\frac{1}{2}} \)