1. **Problem:** Transform the radical expression $$\sqrt[3]{9a^{6}b^{9}}$$ into rational exponents. 2. **Formula:** Recall that $$\sqrt[n]{x} = x^{\frac{1}{n}}$$. For variables with exponents, $$\sqrt[n]{x^{m}} = x^{\frac{m}{n}}$$. 3. **Step-by-step:** - Rewrite each part inside the cube root with rational exponents: $$9 = 9^{1}$$ $$a^{6} = a^{6}$$ $$b^{9} = b^{9}$$ - Apply the cube root as exponent $$\frac{1}{3}$$: $$9^{\frac{1}{3}} a^{6 \times \frac{1}{3}} b^{9 \times \frac{1}{3}} = 9^{\frac{1}{3}} a^{2} b^{3}$$ 4. **Simplify:** - $$9^{\frac{1}{3}}$$ is the cube root of 9, which can be left as is or approximated. --- 1. **Problem:** Transform $$\sqrt[4]{8a^{2}b}$$ into rational exponents. 2. **Formula:** Same as above. 3. **Step-by-step:** - Rewrite inside the fourth root: $$8^{1} a^{2} b^{1}$$ - Apply exponent $$\frac{1}{4}$$: $$8^{\frac{1}{4}} a^{\frac{2}{4}} b^{\frac{1}{4}} = 8^{\frac{1}{4}} a^{\frac{1}{2}} b^{\frac{1}{4}}$$ 4. **Simplify:** - $$a^{\frac{1}{2}}$$ is the square root of $$a$$. --- 1. **Problem:** Transform $$\sqrt[6]{3x^{3}y}$$ into rational exponents. 2. **Formula:** Same as above. 3. **Step-by-step:** - Inside the sixth root: $$3^{1} x^{3} y^{1}$$ - Apply exponent $$\frac{1}{6}$$: $$3^{\frac{1}{6}} x^{\frac{3}{6}} y^{\frac{1}{6}} = 3^{\frac{1}{6}} x^{\frac{1}{2}} y^{\frac{1}{6}}$$ 4. **Simplify:** - $$x^{\frac{1}{2}}$$ is the square root of $$x$$. --- 1. **Problem:** Transform $$\sqrt[4]{25a^{2}b^{3}}$$ into rational exponents. 2. **Formula:** Same as above. 3. **Step-by-step:** - Inside the fourth root: $$25^{1} a^{2} b^{3}$$ - Apply exponent $$\frac{1}{4}$$: $$25^{\frac{1}{4}} a^{\frac{2}{4}} b^{\frac{3}{4}} = 25^{\frac{1}{4}} a^{\frac{1}{2}} b^{\frac{3}{4}}$$ 4. **Simplify:** - $$25^{\frac{1}{4}}$$ is the fourth root of 25. **Final answers:** 1. $$9^{\frac{1}{3}} a^{2} b^{3}$$ 2. $$8^{\frac{1}{4}} a^{\frac{1}{2}} b^{\frac{1}{4}}$$ 3. $$3^{\frac{1}{6}} x^{\frac{1}{2}} y^{\frac{1}{6}}$$ 4. $$25^{\frac{1}{4}} a^{\frac{1}{2}} b^{\frac{3}{4}}$$