Решебник по алгебре 8 класс Мерзляк Задание 574

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\[1)\sqrt{10 + 8\sqrt{2 + \sqrt{9 + 4\sqrt{2}}}} =\]

\[= \sqrt{10 + 8\sqrt{2 + \sqrt{1 + 2\sqrt{8} + 8}}} =\]

\[= \sqrt{10 + 8\sqrt{2 + \sqrt{\left( 1 + \sqrt{8} \right)^{2}}}} =\]

\[= \sqrt{10 + 8\sqrt{2 + 1 + \sqrt{8}}} =\]

\[= \sqrt{10 + 8\sqrt{3 + \sqrt{8}}} =\]

\[= \sqrt{10 + 8\sqrt{1 + 2\sqrt{2} + 2}} =\]

\[= \sqrt{10 + 8\sqrt{\left( 1 + \sqrt{2} \right)^{2}}} =\]

\[= \sqrt{10 + 8 \cdot \left( 1 + \sqrt{2} \right)} =\]

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

\[= \sqrt{\left( 2 + 8\sqrt{2} + 16 \right)} =\]

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

\[2)\ \sqrt{22 + 6\sqrt{3 + \sqrt{13 + \sqrt{48}}}} =\]

\[= \sqrt{22 + 6\sqrt{3 + \sqrt{\left( 1 + \sqrt{12} \right)^{2}}}} =\]

\[= \sqrt{22 + 6\sqrt{3 + 1 + \sqrt{12}}} =\]

\[= \sqrt{22 + 6\sqrt{3 + 2\sqrt{3} + 1}} =\]

\[= \sqrt{22 + 6\sqrt{\left( 1 + \sqrt{3} \right)^{2}}} =\]

\[= \sqrt{22 + 6 + 6\sqrt{3}} =\]

\[= \sqrt{28 + 6\sqrt{3}} =\]

\[= \sqrt{27 + 2\sqrt{27} + 1} =\]

\[= \sqrt{\left( 1 + \sqrt{27} \right)^{2}} = 1 + \sqrt{27} =\]

\[= 1 + 3\sqrt{3}\]