[math]\frac{3}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{3\cdot\sqrt{2}}{\sqrt{4}}=\frac{3\cdot\sqrt{2}}{2}[/math]

[math]\frac{8}{2^{\frac{1}{2}}}=\frac{8}{\sqrt{2}}=\frac{8}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{8\cdot\sqrt{2}}{\sqrt{4}}=\frac{8\cdot\sqrt{2}}{2}=4\cdot\sqrt{2}[/math]

[math]\sqrt{\frac{5}{3}}=\frac{\sqrt{5}}{\sqrt{3}}=\frac{\sqrt{5}}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{5}\cdot\sqrt{3}}{\sqrt{9}}=\frac{\sqrt{5}\cdot\sqrt{3}}{3}=\frac{\sqrt{15}}{3}[/math]

[math]\frac{3\cdot\sqrt{6}}{\sqrt{3}}=\frac{3\cdot\sqrt{6}}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{3\cdot\sqrt{18}}{\sqrt{9}}=\frac{3\cdot\sqrt{18}}{3}=\sqrt{18}[/math][br]Fatorando 18 temos: [math]18=2\cdot9=2\cdot3^2[/math][br]Logo: [math]\sqrt{18}=\sqrt{3^2\cdot2}=3\cdot\sqrt{2}[/math]

[math]\frac{5}{\sqrt[3]{2}}=\frac{5}{\sqrt[3]{2}}\cdot\frac{\sqrt[3]{2^2}}{\sqrt[3]{2^2}}=\frac{5\cdot\sqrt[3]{4}}{\sqrt[3]{2^3}}=\frac{5\cdot\sqrt[3]{4}}{2}[/math]

[math]\frac{7}{\sqrt[4]{3}}=\frac{7}{\sqrt[4]{3}}\cdot\frac{\sqrt[4]{3^3}}{\sqrt[4]{3^3}}=\frac{7\cdot\sqrt[4]{27}}{\sqrt[4]{3^4}}=\frac{7\cdot\sqrt[4]{27}}{3}[/math]

[math]\frac{4-\sqrt{6}}{\sqrt{2}}=\frac{4-\sqrt{6}}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}\cdot\left(4-\sqrt{6}\right)}{\sqrt{4}}=\frac{4\cdot\sqrt{2}-\sqrt{12}}{2}[/math][br]Fatorando o 12 temos: [math]12=4\cdot3=2^2\cdot3[/math][br]Portanto: [math]\frac{4\sqrt{2}-\sqrt{12}}{2}=\frac{4\sqrt{2}-\sqrt{2^2\cdot3}}{2}=\frac{4\sqrt{2}-2\sqrt{3}}{2}=2\sqrt{2}-\sqrt{3}[/math]

[math]\frac{4}{2+\sqrt{2}}=\frac{4}{2+\sqrt{2}}\cdot\frac{2-\sqrt{2}}{2-\sqrt{2}}=\frac{4\left(2-\sqrt{2}\right)}{\left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right)}=\frac{8-4\cdot\sqrt{2}}{2^2-\left(\sqrt{2}\right)^2}[/math][br][math]=\frac{8-4\cdot\sqrt{2}}{4-2}=\frac{8-4\cdot\sqrt{2}}{2}=4-2\sqrt{2}[/math]

[math]\frac{5}{6-\sqrt{3}}=\frac{5}{6-\sqrt{3}}\cdot\frac{6+\sqrt{3}}{6+\sqrt{3}}=\frac{5\cdot\left(6+\sqrt{3}\right)}{\left(6-\sqrt{3}\right)\cdot\left(6+\sqrt{3}\right)}=\frac{30+5\cdot\sqrt{3}}{6^2-\left(\sqrt{3}\right)^2}[/math][br][math]=\frac{30+5\cdot\sqrt{3}}{36-3}=\frac{30+5\cdot\sqrt{3}}{33}[/math]

[math]\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\cdot\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\frac{\left(\sqrt{3}-\sqrt{2}\right)^2}{\left(\sqrt{3}+\sqrt{2}\right)\cdot\left(\sqrt{3}-\sqrt{2}\right)}[/math][br][math]=\frac{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}{\left(\sqrt{3}\right)^2-\left(\sqrt{2}\right)^2}=\frac{3-2\cdot\sqrt{6}+2}{3-2}=3+2-2\cdot\sqrt{6}=5-2\sqrt{6}[/math]