1.220. Упростите выражение: a) 2/(1-2√3) + 2/(1+2√3); б) (√7+√3)/(√7-√3) - (√7-√3)/(√7+√3).

Решение:

a) \( \frac{2}{1 - 2\sqrt{3}} + \frac{2}{1 + 2\sqrt{3}} = \frac{2(1 + 2\sqrt{3}) + 2(1 - 2\sqrt{3})}{(1 - 2\sqrt{3})(1 + 2\sqrt{3})} = \frac{2 + 4\sqrt{3} + 2 - 4\sqrt{3}}{1 - (2\sqrt{3})^2} = \frac{4}{1 - 12} = \frac{4}{-11} = -\frac{4}{11} \)

б) \( \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} - \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}} = \frac{(\sqrt{7} + \sqrt{3})^2 - (\sqrt{7} - \sqrt{3})^2}{(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})} = \frac{(7 + 2\sqrt{21} + 3) - (7 - 2\sqrt{21} + 3)}{7 - 3} = \frac{(10 + 2\sqrt{21}) - (10 - 2\sqrt{21})}{4} = \frac{10 + 2\sqrt{21} - 10 + 2\sqrt{21}}{4} = \frac{4\sqrt{21}}{4} = \sqrt{21} \)

Ответ: a) -4/11; б) √21.