Simplify
\[ \frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} – \frac{7 – 3\sqrt{5}}{3 – \sqrt{5}} \]
Solution:
\[ \frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} \times \frac{3 – \sqrt{5}}{3 – \sqrt{5}} \]
\[ = \frac{(7 + 3\sqrt{5})(3 – \sqrt{5})}{9 – 5} \]
\[ = \frac{21 – 7\sqrt{5} + 9\sqrt{5} – 15}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} \]
\[ \frac{7 – 3\sqrt{5}}{3 – \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} \]
\[ = \frac{(7 – 3\sqrt{5})(3 + \sqrt{5})}{4} \]
\[ = \frac{21 + 7\sqrt{5} – 9\sqrt{5} – 15}{4} = \frac{6 – 2\sqrt{5}}{4} = \frac{3 – \sqrt{5}}{2} \]
\[ \Rightarrow \frac{3 + \sqrt{5}}{2} – \frac{3 – \sqrt{5}}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5} \]