Write the following expression in simplified radical form. \( \sqrt[3]{54 x^{17} w^{3}} \) Assume that all of the variables in the expression represent positive real numbers. \( \square \)

To simplify \( \sqrt[3]{54 x^{17} w^{3}} \), let's break it down step-by-step. First, factor \( 54 \) into its prime factors: \( 54 = 2 \times 3^3 \). Next, we can separate the components under the cube root: \[ \sqrt[3]{54 x^{17} w^{3}} = \sqrt[3]{2 \times 3^3} \times \sqrt[3]{x^{17}} \times \sqrt[3]{w^{3}} \] Now, simplify each part: - \( \sqrt[3]{3^3} = 3 \) - \( \sqrt[3]{x^{17}} = x^{\frac{17}{3}} = x^{5} \sqrt[3]{x^2} \) - \( \sqrt[3]{w^{3}} = w \) Combining these results gives: \[ 3 x^{5} w \sqrt[3]{2 x^2} \] Thus, the simplified radical form is: \[ 3 x^{5} w \sqrt[3]{2 x^2} \]