Simplify
\[ \sqrt{7 + 2\sqrt{6}} – \sqrt{7 – 2\sqrt{6}} \]
Solution:
\[ \sqrt{7 + 2\sqrt{6}} = \sqrt{a} + \sqrt{b}, \quad \sqrt{7 – 2\sqrt{6}} = \sqrt{a} – \sqrt{b} \]
\[ a + b = 7, \quad ab = 6 \Rightarrow a = 6,\ b = 1 \]
\[ \therefore \sqrt{7 + 2\sqrt{6}} = \sqrt{6} + 1,\quad \sqrt{7 – 2\sqrt{6}} = \sqrt{6} – 1 \]
\[ \Rightarrow (\sqrt{6} + 1) – (\sqrt{6} – 1) = 2 \]