[tex]\frac{4}{\sqrt{12}}=\frac{4}{\sqrt{4*3}}=\frac{4}{2\sqrt{3}}*\frac{\sqrt3}{\sqrt{3}}=\frac{4\sqrt3}{2*3}=\frac{2\sqrt3}{3}\\\\\frac{6}{\sqrt8}=\frac{6}{\sqrt{4*2}}=\frac{6}{2\sqrt{2}}*\frac{\sqrt2}{\sqrt2}}=\frac{6\sqrt2}{2*2}=\frac{3\sqrt2}{2}\\\\\frac{9}{\sqrt{27}}=\frac{9}{\sqrt{9*3}}=\frac{9}{3\sqrt3}*\frac{\sqrt3}{\sqrt3}=\frac{9\sqrt3}{3*3}=\sqrt3\\\\\frac{2}{\sqrt{27}}=\frac{2}{\sqrt{9*3}}=\frac{2}{3\sqrt{3}}*\frac{\sqrt3}{\sqrt{3}}=\frac{2\sqrt3}{3*3}=\frac{2\sqrt3}{9}[/tex]
[tex]\frac{12}{\sqrt{50}}=\frac{12}{\sqrt{25*2}}=\frac{12}{5\sqrt{2}}*\frac{\sqrt2}{\sqrt2}=\frac{12\sqrt2}{5*2}=\frac{6\sqrt2}{5}\\\\\frac{6}{\sqrt{75}}=\frac{6}{\sqrt{25*3}}=\frac{6}{5\sqrt{3}}*\frac{\sqrt3}{\sqrt3}=\frac{6\sqrt3}{5*3}=\frac{2\sqrt3}{5}\\\\\frac{\sqrt3}{\sqrt2}=\frac{\sqrt3}{\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{\sqrt6}{2}\\\\\frac{\sqrt5}{\sqrt7}=\frac{\sqrt5}{\sqrt7}*\frac{\sqrt7}{\sqrt7}=\frac{\sqrt{35}}{7}[/tex]
[tex]\frac{\sqrt6}{\sqrt5}=\frac{\sqrt6}{\sqrt5}*\frac{\sqrt5}{\sqrt5}=\frac{\sqrt{30}}{5}\\\\\frac{\sqrt{12}}{\sqrt{175}}=\frac{\sqrt{4*3}}{\sqrt{25*7}}=\frac{2\sqrt{3}}{5\sqrt{7}}*\frac{\sqrt7}{\sqrt7}=\frac{2\sqrt{21}}{5*7}=\frac{2\sqrt{21}}{35}\\\\\frac{\sqrt{125}}{\sqrt{63}}=\frac{\sqrt{25*5}}{\sqrt{9*7}}=\frac{5\sqrt{5}}{3\sqrt{7}}*\frac{\sqrt7}{\sqrt7}=\frac{5\sqrt{35}}{3*7}=\frac{5\sqrt{35}}{21}[/tex]
[tex]\frac{\sqrt{45}}{\sqrt{128}}=\frac{\sqrt{9*5}}{\sqrt{64*2}}=\frac{3\sqrt{5}}{8\sqrt{2}}*\frac{\sqrt2}{\sqrt2}=\frac{3\sqrt{10}}{8*2}=\frac{3\sqrt{10}}{16}\\\\\frac{3-\sqrt2}{\sqrt2}=\frac{3-\sqrt2}{\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{3\sqrt2-2}{2}\\\\\frac{6-\sqrt3}{\sqrt3}=\frac{6-\sqrt3}{\sqrt3}*\frac{\sqrt3}{\sqrt3}=\frac{6\sqrt3-3}{3}=2\sqrt3-1\\\\\frac{\sqrt5-2}{\sqrt5}=\frac{\sqrt5-2}{\sqrt5}*\frac{\sqrt5}{\sqrt5}=\frac{5-2\sqrt5}{5}[/tex]
[tex]\frac{\sqrt6+1}{\sqrt6}=\frac{\sqrt6+1}{\sqrt6}*\frac{\sqrt6}{\sqrt6}=\frac{6+\sqrt6}{6}\\\\\frac{3+\sqrt2}{5\sqrt2}=\frac{3+\sqrt2}{5\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{3\sqrt2+2}{5*2}=\frac{3\sqrt2+2}{10}\\\\\frac{\sqrt2+5}{2\sqrt2}=\frac{\sqrt2+5}{2\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{2+5\sqrt2}{2*2}=\frac{2+5\sqrt2}{4}[/tex]
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[tex]\frac{4}{\sqrt{12}}=\frac{4}{\sqrt{4*3}}=\frac{4}{2\sqrt{3}}*\frac{\sqrt3}{\sqrt{3}}=\frac{4\sqrt3}{2*3}=\frac{2\sqrt3}{3}\\\\\frac{6}{\sqrt8}=\frac{6}{\sqrt{4*2}}=\frac{6}{2\sqrt{2}}*\frac{\sqrt2}{\sqrt2}}=\frac{6\sqrt2}{2*2}=\frac{3\sqrt2}{2}\\\\\frac{9}{\sqrt{27}}=\frac{9}{\sqrt{9*3}}=\frac{9}{3\sqrt3}*\frac{\sqrt3}{\sqrt3}=\frac{9\sqrt3}{3*3}=\sqrt3\\\\\frac{2}{\sqrt{27}}=\frac{2}{\sqrt{9*3}}=\frac{2}{3\sqrt{3}}*\frac{\sqrt3}{\sqrt{3}}=\frac{2\sqrt3}{3*3}=\frac{2\sqrt3}{9}[/tex]
[tex]\frac{12}{\sqrt{50}}=\frac{12}{\sqrt{25*2}}=\frac{12}{5\sqrt{2}}*\frac{\sqrt2}{\sqrt2}=\frac{12\sqrt2}{5*2}=\frac{6\sqrt2}{5}\\\\\frac{6}{\sqrt{75}}=\frac{6}{\sqrt{25*3}}=\frac{6}{5\sqrt{3}}*\frac{\sqrt3}{\sqrt3}=\frac{6\sqrt3}{5*3}=\frac{2\sqrt3}{5}\\\\\frac{\sqrt3}{\sqrt2}=\frac{\sqrt3}{\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{\sqrt6}{2}\\\\\frac{\sqrt5}{\sqrt7}=\frac{\sqrt5}{\sqrt7}*\frac{\sqrt7}{\sqrt7}=\frac{\sqrt{35}}{7}[/tex]
[tex]\frac{\sqrt6}{\sqrt5}=\frac{\sqrt6}{\sqrt5}*\frac{\sqrt5}{\sqrt5}=\frac{\sqrt{30}}{5}\\\\\frac{\sqrt{12}}{\sqrt{175}}=\frac{\sqrt{4*3}}{\sqrt{25*7}}=\frac{2\sqrt{3}}{5\sqrt{7}}*\frac{\sqrt7}{\sqrt7}=\frac{2\sqrt{21}}{5*7}=\frac{2\sqrt{21}}{35}\\\\\frac{\sqrt{125}}{\sqrt{63}}=\frac{\sqrt{25*5}}{\sqrt{9*7}}=\frac{5\sqrt{5}}{3\sqrt{7}}*\frac{\sqrt7}{\sqrt7}=\frac{5\sqrt{35}}{3*7}=\frac{5\sqrt{35}}{21}[/tex]
[tex]\frac{\sqrt{45}}{\sqrt{128}}=\frac{\sqrt{9*5}}{\sqrt{64*2}}=\frac{3\sqrt{5}}{8\sqrt{2}}*\frac{\sqrt2}{\sqrt2}=\frac{3\sqrt{10}}{8*2}=\frac{3\sqrt{10}}{16}\\\\\frac{3-\sqrt2}{\sqrt2}=\frac{3-\sqrt2}{\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{3\sqrt2-2}{2}\\\\\frac{6-\sqrt3}{\sqrt3}=\frac{6-\sqrt3}{\sqrt3}*\frac{\sqrt3}{\sqrt3}=\frac{6\sqrt3-3}{3}=2\sqrt3-1\\\\\frac{\sqrt5-2}{\sqrt5}=\frac{\sqrt5-2}{\sqrt5}*\frac{\sqrt5}{\sqrt5}=\frac{5-2\sqrt5}{5}[/tex]
[tex]\frac{\sqrt6+1}{\sqrt6}=\frac{\sqrt6+1}{\sqrt6}*\frac{\sqrt6}{\sqrt6}=\frac{6+\sqrt6}{6}\\\\\frac{3+\sqrt2}{5\sqrt2}=\frac{3+\sqrt2}{5\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{3\sqrt2+2}{5*2}=\frac{3\sqrt2+2}{10}\\\\\frac{\sqrt2+5}{2\sqrt2}=\frac{\sqrt2+5}{2\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{2+5\sqrt2}{2*2}=\frac{2+5\sqrt2}{4}[/tex]