Simplify
\[ \frac{3\sqrt{2} – 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}} + \frac{\sqrt{12}}{\sqrt{3} – \sqrt{2}} \]
Solution:
\[ \frac{3\sqrt{2} – 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}} \times \frac{3\sqrt{2} – 2\sqrt{3}}{3\sqrt{2} – 2\sqrt{3}} \]
\[ = \frac{(3\sqrt{2} – 2\sqrt{3})^2}{(3\sqrt{2})^2 – (2\sqrt{3})^2} \]
\[ = \frac{18 – 12\sqrt{6} + 12}{18 – 12} \]
\[ = \frac{30 – 12\sqrt{6}}{6} = 5 – 2\sqrt{6} \]
\[ \frac{\sqrt{12}}{\sqrt{3} – \sqrt{2}} = \frac{2\sqrt{3}}{\sqrt{3} – \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} \]
\[ = \frac{2\sqrt{3}(\sqrt{3} + \sqrt{2})}{3 – 2} \]
\[ = 2\sqrt{3}(\sqrt{3} + \sqrt{2}) \]
\[ = 6 + 2\sqrt{6} \]
\[ \Rightarrow (5 – 2\sqrt{6}) + (6 + 2\sqrt{6}) = 11 \]