write in exponential form \( \sqrt[3]{x^{2}} \) \( \sqrt[4]{x^{5} 4^{6}} \) \( 5 \sqrt[4]{x^{15}} \)

Certainly! Let's rewrite each expression in exponential form using fractional exponents. 1. **\(\sqrt[3]{x^{2}}\)** The cube root can be expressed as an exponent of \( \frac{1}{3} \): \[ \sqrt[3]{x^{2}} = x^{\frac{2}{3}} \] 2. **\(\sqrt[4]{x^{5} \cdot 4^{6}}\)** The fourth root can be expressed as an exponent of \( \frac{1}{4} \). Apply the exponent to each factor inside the root: \[ \sqrt[4]{x^{5} \cdot 4^{6}} = (x^{5})^{\frac{1}{4}} \cdot (4^{6})^{\frac{1}{4}} = x^{\frac{5}{4}} \cdot 4^{\frac{6}{4}} = x^{\frac{5}{4}} \cdot 4^{\frac{3}{2}} \] Alternatively, it can also be written as: \[ \sqrt[4]{x^{5} \cdot 4^{6}} = (x^{5} \cdot 4^{6})^{\frac{1}{4}} \] 3. **\(5 \sqrt[4]{x^{15}}\)** The fourth root of \(x^{15}\) is expressed as \(x^{\frac{15}{4}}\), and it's multiplied by 5: \[ 5 \sqrt[4]{x^{15}} = 5x^{\frac{15}{4}} \] **Summary:** \[ \begin{align*} \sqrt[3]{x^{2}} &= x^{\frac{2}{3}} \\ \sqrt[4]{x^{5} \cdot 4^{6}} &= x^{\frac{5}{4}} \cdot 4^{\frac{3}{2}} \\ 5 \sqrt[4]{x^{15}} &= 5x^{\frac{15}{4}} \end{align*} \]