Упростите выражение:
1) $\sqrt{18 + 8\sqrt{2}}$;
2) $\sqrt{38 - 12\sqrt{2}}$;
3) $\sqrt{16 + 6\sqrt{7}} + \sqrt{23 - 8\sqrt{7}}$;
4) $\sqrt{26 - 6\sqrt{17}} - \sqrt{66 - 14\sqrt{17}}$;
5) $\sqrt{46 + 10\sqrt{21}} + \sqrt{46 - 10\sqrt{21}}$.
$\sqrt{18 + 8\sqrt{2}} = \sqrt{16 + 2 + 2 * 4 * \sqrt{2}} = \sqrt{4^2 + 2 * 4 * \sqrt{2} + (\sqrt{2})^2} = \sqrt{(4 + \sqrt{2})^2} = |4 + \sqrt{2}| = 4 + \sqrt{2}$
$\sqrt{38 - 12\sqrt{2}} = \sqrt{36 + 2 - 2 * 6 * \sqrt{2}} = \sqrt{6^2 - 2 * 6 * \sqrt{2} + (\sqrt{2})^2} = \sqrt{(6 - \sqrt{2})^2} = |6 - \sqrt{2}| = 6 - \sqrt{2}$
$\sqrt{16 + 6\sqrt{7}} + \sqrt{23 - 8\sqrt{7}} = \sqrt{9 + 7 + 2 * 3 * \sqrt{7}} + \sqrt{16 + 7 - 2 * 4 * \sqrt{7}} = \sqrt{3^2 + 2 * 3 * \sqrt{2} + (\sqrt{7})^2} + \sqrt{4^2 - 2 * 4 * \sqrt{7} + (\sqrt{7})^2} = \sqrt{(3 + \sqrt{7})^2} + \sqrt{(4 - \sqrt{7})^2} = |3 + \sqrt{7}| + |4 - \sqrt{7}| = 3 + \sqrt{7} + 4 - \sqrt{7} = 7$
$\sqrt{26 - 6\sqrt{17}} - \sqrt{66 - 14\sqrt{17}} = \sqrt{9 + 17 - 2 * 3 * \sqrt{17}} - \sqrt{49 + 17 - 2 * 7 * \sqrt{17}} = \sqrt{3^2 - 2 * 3 * \sqrt{17} + (\sqrt{17})^2} - \sqrt{7^2 - 2 * 7 * \sqrt{17} + (\sqrt{17})^2} = \sqrt{(3 - \sqrt{17})^2} - \sqrt{(7 - \sqrt{17})^2} = |3 - \sqrt{17}| - |7 - \sqrt{17}| = -3 + \sqrt{17} - 7 + \sqrt{17} = 2\sqrt{17} - 10$
$\sqrt{46 + 10\sqrt{21}} + \sqrt{46 - 10\sqrt{21}} = \sqrt{25 + 21 + 2 * 5 * \sqrt{21}} + \sqrt{25 + 21 - 2 * 5\sqrt{21}} = \sqrt{5^2 + 2 * 5 * \sqrt{21} + (\sqrt{21})^2} + \sqrt{5^2 - 2 * 5\sqrt{21} + (\sqrt{21})^2} = \sqrt{(5 + \sqrt{21})^2} + \sqrt{(5 - \sqrt{21})^2} = |5 + \sqrt{21}| + |5 - \sqrt{21}| = 5 + \sqrt{21} + 5 - \sqrt{21} = 10$
Пожауйста, оцените решение
