05. Упростите выражение: a) 15√2/5 - √160; b) 6√1 1/3 - √27; б) √135 + 10√0,6; г) 0,5√24 + 10√3/8.

Решение:

а)

• \[15 \sqrt{\frac{2}{5}} - \sqrt{160} = 15 \frac{\sqrt{2}}{\sqrt{5}} - \sqrt{16 \times 10} = 15 \frac{\sqrt{2} \times \sqrt{5}}{5} - 4 \sqrt{10} = 3 \sqrt{10} - 4 \sqrt{10} = -\sqrt{10}\]

б)

• \[\sqrt{135} + 10\sqrt{0,6} = \sqrt{9 \times 15} + 10\sqrt{\frac{6}{10}} = 3\sqrt{15} + 10\sqrt{\frac{3}{5}} = 3\sqrt{15} + 10 \frac{\sqrt{3}}{\sqrt{5}} = 3\sqrt{15} + 10 \frac{\sqrt{3} \times \sqrt{5}}{5} = 3\sqrt{15} + 2\sqrt{15} = 5\sqrt{15}\]

в)

• \[6 \sqrt{1\frac{1}{3}} - \sqrt{27} = 6 \sqrt{\frac{4}{3}} - \sqrt{9 \times 3} = 6 \frac{\sqrt{4}}{\sqrt{3}} - 3\sqrt{3} = 6 \frac{2}{\sqrt{3}} - 3\sqrt{3} = \frac{12}{\sqrt{3}} - 3\sqrt{3} = \frac{12\sqrt{3}}{3} - 3\sqrt{3} = 4\sqrt{3} - 3\sqrt{3} = \sqrt{3}\]

г)

• \[0,5\sqrt{24} + 10\sqrt{\frac{3}{8}} = \frac{1}{2} \sqrt{4 \times 6} + 10 \frac{\sqrt{3}}{\sqrt{8}} = \frac{1}{2} \times 2\sqrt{6} + 10 \frac{\sqrt{3}}{2\sqrt{2}} = \sqrt{6} + 5 \frac{\sqrt{3}}{ \sqrt{2}} = \sqrt{6} + 5 \frac{\sqrt{3} \times \sqrt{2}}{2} = \sqrt{6} + \frac{5\sqrt{6}}{2} = \sqrt{6} \left(1 + \frac{5}{2}\right) = \sqrt{6} \left(\frac{2}{2} + \frac{5}{2}\right) = \frac{7}{2}\sqrt{6}\]

Ответ:

• а) $$-\sqrt{10}$$
• б) $$5\sqrt{15}$$
• в) $$\sqrt{3}$$
• г) $$\frac{7}{2}\sqrt{6}$$