Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 454

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\[1)\ \sqrt{9 + \sqrt{17}} \bullet \sqrt{9 - \sqrt{17}} =\]

\[= \sqrt{\left( 9 + \sqrt{17} \right)\left( 9 - \sqrt{17} \right)} =\]

\[= \sqrt{(9^{2} - \left( \sqrt{17} \right)^{2}} =\]

\[= \sqrt{81 - 17} = \sqrt{64} = 8;\]

\[2)\ \left( \sqrt{3 + \sqrt{5}} - \sqrt{3 - \sqrt{5}} \right)^{2} =\]

\[= \left( 3 + \sqrt{5} \right) - 2\sqrt{3 + \sqrt{5}} \bullet\]

\[\bullet \sqrt{3 - \sqrt{5}} + \left( 3 - \sqrt{5} \right) =\]

\[= 3 + \sqrt{5} + 3 - \sqrt{5} -\]

\[- 2\sqrt{\left( 3 + \sqrt{5} \right)\left( 3 - \sqrt{5} \right)} =\]

\[= 6 - 2\sqrt{9 - 5} = 6 - 2\sqrt{4} =\]

\[= 6 - 2 \bullet 2 = 6 - 4 = 2;\]

\[3)\ \left( \sqrt{5 + \sqrt{21}} + \sqrt{5 - \sqrt{21}} \right)^{2} =\]

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

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

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

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

\[= 10 + 2\sqrt{25 - 21} =\]

\[= 10 + 2\sqrt{4} = 10 + 2 \bullet 2 =\]

\[= 10 + 4 = 14.\]