1. $4^{\frac{2}{3}} = \sqrt[3]{4^2}$
2. $a^{\frac{4}{5}} = \sqrt[5]{a^4}$
3. $x^{\frac{1}{2}}y^{\frac{3}{4}}= \sqrt{x} \cdot \sqrt[4]{y^3}$
4. $(4^{\frac{2}{3}})(4^{\frac{1}{3}})= 4^{\frac{3}{3}}=4^1=4$
5. $(2^{\frac{4}{5}})^{\frac{1}{2}}= 2^\frac{4}{10}=2^{\frac{2}{5}}=\sqrt[5]{2^2}$
6. $1^{\frac{1}{2}}3^{\frac{3}{4}}= 1(\sqrt[4]{3^3}=\sqrt[4]{3^3}$
7. $(x^{\frac{2}{5}})^{\frac{2}{3}}=x^{\frac{4}{15}}=\sqrt[15]{x^4}$
8. $(3^2)^{\frac{2}{3}}= 3^{\frac{4}{3}}=\sqrt[3]{3^4}=3\sqrt[3]{3}$
9. $\dfrac{x^{\frac{1}{4}}}{x^{\frac{1}{2}}}=x^{-\frac{1}{4}}=\dfrac{1}{\sqrt[4]{x}}$
10. $\dfrac{x^{\frac{1}{4}}y^{\frac{2}{3}}}{x^{\frac{1}{8}}y^{\frac{1}{3}}}=\dfrac{y^{\frac{1}{3}}}{x^{\frac{1}{8}}}=\dfrac{\sqrt[3]{y}}{\sqrt[8]{x}}$
11. $\dfrac{2^{\frac{5}{6}}}{2^{\frac{2}{3}}}=2^{\frac{1}{6}}=\sqrt[6]{2}$
12. $\dfrac{3^{\frac{2}{3}}}{3^{\frac{4}{3}}}=3^{-\frac{2}{3}}=\dfrac{1}{\sqrt[3]{3^2}}$