1. **Problem:** Rewrite each radical as an expression using rational exponents. 2. **Formula:** The nth root of a number $a^m$ can be written as $a^{\frac{m}{n}}$. 3. **Solutions:** - a) $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ - b) $\sqrt{x} = x^{\frac{1}{2}}$ - c) $\sqrt[3]{x} = x^{\frac{1}{3}}$ - d) $\sqrt[4]{x^3} = x^{\frac{3}{4}}$ - e) $(\sqrt[5]{x})^3 = (x^{\frac{1}{5}})^3 = x^{\frac{3}{5}}$ - f) $\frac{1}{\sqrt{x^5}} = x^{-\frac{5}{2}}$ - g) $(\frac{1}{\sqrt[3]{x}})^{11} = (x^{-\frac{1}{3}})^{11} = x^{-\frac{11}{3}}$ - h) $\sqrt{\frac{x}{y^3}} = \left(\frac{x}{y^3}\right)^{\frac{1}{2}} = \frac{x^{\frac{1}{2}}}{y^{\frac{3}{2}}}$ **Final answers for 1:** a) $a^{\frac{m}{n}}$, b) $x^{\frac{1}{2}}$, c) $x^{\frac{1}{3}}$, d) $x^{\frac{3}{4}}$, e) $x^{\frac{3}{5}}$, f) $x^{-\frac{5}{2}}$, g) $x^{-\frac{11}{3}}$, h) $\frac{x^{\frac{1}{2}}}{y^{\frac{3}{2}}}$