Simplify
\[ \frac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} – \frac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} – \frac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} \]
Solution:
\[ \frac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} \times \frac{\sqrt{10} – \sqrt{3}}{\sqrt{10} – \sqrt{3}} = \frac{7\sqrt{3}(\sqrt{10} – \sqrt{3})}{10 – 3} = \sqrt{3}(\sqrt{10} – \sqrt{3}) \]
\[ = \sqrt{30} – 3 \]
\[ \frac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} \times \frac{\sqrt{6} – \sqrt{5}}{\sqrt{6} – \sqrt{5}} = 2\sqrt{5}(\sqrt{6} – \sqrt{5}) \]
\[ = 2\sqrt{30} – 10 \]
\[ \frac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} \times \frac{\sqrt{15} – 3\sqrt{2}}{\sqrt{15} – 3\sqrt{2}} = \frac{3\sqrt{2}(\sqrt{15} – 3\sqrt{2})}{15 – 18} \]
\[ = -\sqrt{2}(\sqrt{15} – 3\sqrt{2}) \]
\[ = -\sqrt{30} + 6 \]
\[ \Rightarrow (\sqrt{30} – 3) – (2\sqrt{30} – 10) – (-\sqrt{30} + 6) \]
\[ = \sqrt{30} – 3 – 2\sqrt{30} + 10 + \sqrt{30} – 6 \]
\[ = (\sqrt{30} – 2\sqrt{30} + \sqrt{30}) + ( -3 + 10 – 6 ) = 0 + 1 = 1 \]