Express \( 75^{\frac{1}{3}} \) in simplest radical form.

To express \( 75^{\frac{1}{3}} \) in simplest radical form, we start by factoring 75 into its prime factors: \[ 75 = 3 \times 5^2 \] Now, we can rewrite \( 75^{\frac{1}{3}} \) using this factorization: \[ 75^{\frac{1}{3}} = (3 \times 5^2)^{\frac{1}{3}} = 3^{\frac{1}{3}} \times (5^2)^{\frac{1}{3}} \] The term \( (5^2)^{\frac{1}{3}} \) simplifies to \( 5^{\frac{2}{3}} \). Thus, we have: \[ 75^{\frac{1}{3}} = 3^{\frac{1}{3}} \times 5^{\frac{2}{3}} \] Next, we can express each term in radical form. The expression \( 3^{\frac{1}{3}} \) corresponds to \( \sqrt[3]{3} \) and \( 5^{\frac{2}{3}} \) corresponds to \( 5^{\frac{2}{3}} = \sqrt[3]{5^2} = \frac{5^2}{\sqrt[3]{5}} \). Putting it all together, we can rewrite: \[ 75^{\frac{1}{3}} = \frac{5^2}{\sqrt[3]{5}} \sqrt[3]{3} = \sqrt[3]{75} \] However, if we want to simplify it directly in radical form, we can express: \[ 75^{\frac{1}{3}} = \sqrt[3]{75} \] Thus, the simplest radical form of \( 75^{\frac{1}{3}} \) is: \[ \sqrt[3]{75} \]