Решебник по алгебре 9 класс Мерзляк Задание 86

\[\boxed{\text{86\ (86).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\ \sqrt{10} + \sqrt{6}\ и\ \sqrt{11} + \sqrt{6}\]

\[\left( \sqrt{10} + \sqrt{6} \right)^{2} = 10 + 2\sqrt{60} + 6 =\]

\[= 16 + 2\sqrt{60}\]

\[\left( \sqrt{11} + \sqrt{5} \right)^{2} = 11 + 2\sqrt{55} + 5 =\]

\[= 16 + 2\sqrt{55}\]

\[\sqrt{60} > \sqrt{55}:\]

\[6 + 2\sqrt{60} > 16 + 2\sqrt{55}.\]

\[Отсюда:\]

\[\sqrt{10} + \sqrt{6} > \sqrt{11} + \sqrt{5}.\]

\[2)\ 2 + \sqrt{11}\ \ и\ \sqrt{5} + \sqrt{10}\]

\[\left( 2 + \sqrt{11} \right)^{2} = 4 + 4\sqrt{11} + 11 =\]

\[= 15 + 4\sqrt{11} = 15 + \sqrt{176}\]

\[\left( \sqrt{5} + \sqrt{10} \right)^{2} = 5 + 2\sqrt{50} + 10 =\]

\[= 15 + 2\sqrt{50} = 15 + \sqrt{200}\]

\[\sqrt{200} > \sqrt{176}:\]

\[2 + \sqrt{11} < \sqrt{5} + \sqrt{10}.\]

\[3)\ \sqrt{15} - \sqrt{5}\ и\ \ \sqrt{2}\]

\[\left( \sqrt{15} - \sqrt{5} \right)^{2} = 15 - 2\sqrt{75} + 5 =\]

\[= 20 - 10\sqrt{3}\]

\[\left( \sqrt{2} \right)^{2} = 2\]

\[20 - 10\sqrt{3} > 2.\]

\[Отсюда:\]

\[\sqrt{15} - \sqrt{5} > \sqrt{2}\]

\[4)\ \sqrt{21} + \sqrt{20}\ и\ 9\]

\[\left( \sqrt{21} + \sqrt{20} \right)^{2} = 21 + 20 +\]

\[+ 2\sqrt{420} = 41 + 4\sqrt{105}\]

\[9^{2} = 81\]

\[41 + 4\sqrt{105} > 81\]

\[\sqrt{105} > 10:\]

\[\sqrt{21} + \sqrt{20} > 9.\]