1.217. Упростите выражение: a) √7+21/√7; б) (18-5√3)/√3; в) (√6/√2 + √2)·√2; г) (2/√18) : (3/2).

Решение:

a) \( \sqrt{7} + \frac{21}{\sqrt{7}} = \sqrt{7} + \frac{21\sqrt{7}}{7} = \sqrt{7} + 3\sqrt{7} = 4\sqrt{7} \)

б) \( \frac{18 - 5\sqrt{3}}{\sqrt{3}} = \frac{18}{\sqrt{3}} - \frac{5\sqrt{3}}{\sqrt{3}} = \frac{18\sqrt{3}}{3} - 5 = 6\sqrt{3} - 5 \)

в) \( (\frac{\sqrt{6}}{\sqrt{2}} + \sqrt{2}) \cdot \sqrt{2} = (\sqrt{\frac{6}{2}} + \sqrt{2}) \cdot \sqrt{2} = (\sqrt{3} + \sqrt{2}) \cdot \sqrt{2} = \sqrt{3} \cdot \sqrt{2} + \sqrt{2} \cdot \sqrt{2} = \sqrt{6} + 2 \)

г) \( \frac{2}{\sqrt{18}} : \frac{3}{2} = \frac{2}{\sqrt{18}} \cdot \frac{2}{3} = \frac{4}{3\sqrt{18}} = \frac{4}{3\sqrt{9 \cdot 2}} = \frac{4}{3 \cdot 3\sqrt{2}} = \frac{4}{9\sqrt{2}} = \frac{4\sqrt{2}}{9 \cdot 2} = \frac{4\sqrt{2}}{18} = \frac{2\sqrt{2}}{9} \)

Ответ: а) \( 4\sqrt{7} \); б) \( 6\sqrt{3} - 5 \); в) \( \sqrt{6} + 2 \); г) \( \frac{2\sqrt{2}}{9} \).