Express in simplest radical form. \[ \left(8 x^{5}\right)^{\frac{2}{3}} \]

To express \((8 x^{5})^{\frac{2}{3}}\) in simplest radical form, first break it down: 1. Rewrite the expression: \[ (8)^{\frac{2}{3}} \cdot (x^{5})^{\frac{2}{3}} \] 2. Calculate \((8)^{\frac{2}{3}}\): - The cube root of 8 is \(2\) (since \(2^3 = 8\)). - Then square it: \(2^2 = 4\). Thus, \((8)^{\frac{2}{3}} = 4\). 3. Calculate \((x^{5})^{\frac{2}{3}}\): - Multiply the exponents: \(x^{\frac{5 \cdot 2}{3}} = x^{\frac{10}{3}}\). 4. Now we can put it all together: \[ (8 x^{5})^{\frac{2}{3}} = 4 x^{\frac{10}{3}} \] 5. To express \(x^{\frac{10}{3}}\) in radical form, write it as: \[ x^{\frac{10}{3}} = x^{3} \cdot x^{\frac{1}{3}} = x^{3} \sqrt[3]{x} \] Combining everything, we have: \[ \left(8 x^{5}\right)^{\frac{2}{3}} = 4 x^{3} \sqrt[3]{x} \] Thus, the simplest radical form is: \[ 4 x^{3} \sqrt[3]{x} \]