Решебник по алгебре 8 класс Мерзляк Задание 897

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\[1)\ \sqrt{18 + 8\sqrt{2}} =\]

\[= \sqrt{2 + 16 + 8\sqrt{2}} =\]

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

\[2)\ \sqrt{38 - 12\sqrt{2}} =\]

\[= \sqrt{2 + 36 - 12\sqrt{2}} =\]

\[= \sqrt{\left( \sqrt{2} - 6 \right)^{2}} = \left| \sqrt{2} - 6 \right| =\]

\[= 6 - \sqrt{2}\]

\[3)\sqrt{16 + 6\sqrt{7}} + \sqrt{23 - 8\sqrt{7}} =\]

\[= \sqrt{\left( \sqrt{7} + 3 \right)^{2}} + \sqrt{\left( \sqrt{7} - 4 \right)^{2}} =\]

\[= \sqrt{7} + 3 + 4 - \sqrt{7} = 7\]

\[= \sqrt{\left( \sqrt{17} - 3 \right)^{2}} - \sqrt{\left( \sqrt{17} - 7 \right)^{2}} =\]

\[= \sqrt{17} - 3 - 7 + \sqrt{17} =\]

\[= 2\sqrt{17} - 10\]

\[= \sqrt{\left( \sqrt{21} + 5 \right)^{2}} + \sqrt{\left( \sqrt{21} - 5 \right)^{2}} =\]

\[= \sqrt{21} + 5 + 5 - \sqrt{21} = 10\]