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 find the expression equivalent to \( \sqrt{75 x} \), first, simplify \( \sqrt{75} \). Since \( 75 = 25 \times 3 \), we have: \[ \sqrt{75 x} = \sqrt{25 \cdot 3 \cdot x} = \sqrt{25} \cdot \sqrt{3} \cdot \sqrt{x} = 5 \sqrt{3x}. \] Now, let's analyze the given choices: - \( (5x)^{\frac{1}{2}} = \sqrt{5x} \) (not equivalent) - \( (5x)^{\frac{75}{2}} \) (not equivalent) - \( 5(3x)^{\frac{1}{2}} = 5 \sqrt{3x} \) (this one matches!) - \( 5(25x)^{\frac{1}{3}} \) (not equivalent) Thus, the equivalent expression is \( 5(3x)^{\frac{1}{2}} \).