1. $\dfrac{4+2\sqrt{3}}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}\Rightarrow \dfrac{4\sqrt{3}+2(3)}{3}\Rightarrow \dfrac{4\sqrt{3}+6}{3}$
2. $\dfrac{-4+\sqrt{3}}{4\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}\Rightarrow \dfrac{-4\sqrt{3}+3}{4(3)}\Rightarrow \dfrac{-4\sqrt{3}+3}{12}$
3. $\dfrac{4+2\sqrt{3}}{5\sqrt{6}}\cdot \dfrac{\sqrt{6}}{\sqrt{6}}\Rightarrow \dfrac{4\sqrt{6}+2\sqrt{18}}{5(6)}\Rightarrow \dfrac{4\sqrt{6}+6\sqrt{2}}{30}\Rightarrow \dfrac{2\sqrt{6}+3\sqrt{2}}{15}$
4. $\dfrac{2\sqrt{3}-2}{2\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}\Rightarrow \dfrac{2(3)-2\sqrt{3}}{2(3)}\Rightarrow \dfrac{6-2\sqrt{3}}{6}\Rightarrow \dfrac{3-\sqrt{3}}{3}$
5. $\dfrac{2-5\sqrt{5}}{4\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}\Rightarrow \dfrac{2\sqrt{3}-5\sqrt{15}}{4(3)}\Rightarrow \dfrac{2\sqrt{3}-5\sqrt{15}}{12}$
6. $\dfrac{\sqrt{5}+4}{4\sqrt{5}}\cdot \dfrac{\sqrt{5}}{\sqrt{5}}\Rightarrow \dfrac{5+4\sqrt{5}}{4(5)}\Rightarrow \dfrac{5+4\sqrt{5}}{20}$
7. $\dfrac{\sqrt{2}-3\sqrt{3}}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}\Rightarrow \dfrac{\sqrt{6}-3(3)}{3}\Rightarrow \dfrac{\sqrt{6}-9}{3}$
8. $\dfrac{\sqrt{5}-\sqrt{2}}{3\sqrt{6}}\cdot \dfrac{\sqrt{6}}{\sqrt{6}}\Rightarrow \dfrac{\sqrt{30}-\sqrt{12}}{3(6)}\Rightarrow \dfrac{\sqrt{30}-2\sqrt{3}}{18}$
9. $\dfrac{5}{3\sqrt{5}+\sqrt{2}}\cdot \dfrac{3\sqrt{5}-\sqrt{2}}{3\sqrt{5}-\sqrt{2}}\Rightarrow \dfrac{15\sqrt{5}-5\sqrt{2}}{9(5)-2}\Rightarrow \dfrac{15\sqrt{5}-5\sqrt{2}}{43}$
10. $\dfrac{5}{\sqrt{3}+4\sqrt{5}}\cdot \dfrac{\sqrt{3}-4\sqrt{5}}{\sqrt{3}-4\sqrt{5}}\Rightarrow \dfrac{5\sqrt{3}-20\sqrt{5}}{3-16(5)}\Rightarrow \dfrac{5\sqrt{3}-20\sqrt{5}}{-77}\Rightarrow \dfrac{20\sqrt{5}-5\sqrt{3}}{77}$
11. $\dfrac{2}{5+\sqrt{2}}\cdot \dfrac{5-\sqrt{2}}{5-\sqrt{2}}\Rightarrow \dfrac{10-2\sqrt{2}}{25-2}\Rightarrow \dfrac{10-2\sqrt{2}}{23}$
12. $\dfrac{5}{2\sqrt{3}-\sqrt{2}}\cdot \dfrac{2\sqrt{3}+\sqrt{2}}{2\sqrt{3}+\sqrt{2}}\Rightarrow \dfrac{10\sqrt{3}+5\sqrt{2}}{4(3)-2}\Rightarrow \dfrac{10\sqrt{3}+5\sqrt{2}}{10}\Rightarrow \dfrac{2\sqrt{3}+\sqrt{2}}{2}$
13. $\dfrac{3}{4-\sqrt{3}}\cdot \dfrac{4+\sqrt{3}}{4+\sqrt{3}}\Rightarrow \dfrac{12+3\sqrt{3}}{16-3}\Rightarrow \dfrac{12+3\sqrt{3}}{13}$
14. $\dfrac{4}{\sqrt{2}-2}\cdot \dfrac{\sqrt{2}+2}{\sqrt{2}+2}\Rightarrow \dfrac{4\sqrt{2}+8}{2-4}\Rightarrow \dfrac{4\sqrt{2}+8}{-2}\Rightarrow -2\sqrt{2}-4$
15. $\dfrac{4}{3+\sqrt{5}}\cdot \dfrac{3-\sqrt{5}}{3-\sqrt{5}}\Rightarrow \dfrac{12-4\sqrt{5}}{9-5}\Rightarrow \dfrac{12-4\sqrt{5}}{4}\Rightarrow 3-\sqrt{5}$
16. $\dfrac{2}{\sqrt{5}+2\sqrt{3}}\cdot \dfrac{\sqrt{5}-2\sqrt{3}}{\sqrt{5}-2\sqrt{3}}\Rightarrow \dfrac{2\sqrt{5}-4\sqrt{3}}{5-4(3)}\Rightarrow \dfrac{2\sqrt{5}-4\sqrt{3}}{-7}$
17. $\dfrac{-3+2\sqrt{3}}{\sqrt{3}+2}\cdot \dfrac{\sqrt{3}-2}{\sqrt{3}-2}\Rightarrow \dfrac{-3\sqrt{3}+6+2(3)-4\sqrt{3}}{3-4}\Rightarrow \dfrac{12-7\sqrt{3}}{-1}\Rightarrow$
$\phantom{a}$
$-12+7\sqrt{3}$
18. $\dfrac{4+\sqrt{5}}{2+2\sqrt{5}}\cdot \dfrac{2-2\sqrt{5}}{2-2\sqrt{5}}\Rightarrow \dfrac{8-8\sqrt{5}+2\sqrt{5}-2(5)}{4-4(5)}\Rightarrow \dfrac{-2-6\sqrt{5}}{-16}\Rightarrow \dfrac{1+3\sqrt{5}}{8}$
19. $\dfrac{2-\sqrt{3}}{1+\sqrt{2}}\cdot \dfrac{1-\sqrt{2}}{1-\sqrt{2}}\Rightarrow \dfrac{2-2\sqrt{2}-\sqrt{3}+\sqrt{6}}{1-2}\Rightarrow \dfrac{2-2\sqrt{2}-\sqrt{3}+\sqrt{6}}{-1}$
$\phantom{a}$
$\Rightarrow2\sqrt{2}+\sqrt{3}-\sqrt{6}-2$
20. $\dfrac{-1+\sqrt{3}}{\sqrt{3}-1}\cdot \dfrac{\sqrt{3}+1}{\sqrt{3}+1}\Rightarrow \dfrac{-\sqrt{3}-1+3+\sqrt{3}}{3-1}\Rightarrow \dfrac{2}{2}\Rightarrow 1$