\[\begin{array}{l} {\text{Докажите}}{\text{, что}} \hfill \\ \sqrt {12 - 2\sqrt {11} } - \sqrt {15 - 4\sqrt {11} } \hfill \\ {\text{ - целое число}}{\text{.}} \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Докажите равенство:}} \hfill \\ \sqrt {a \pm \sqrt b } = \sqrt {\frac{{a + \sqrt {{a^2} - b} }}{2}} \pm \sqrt {\frac{{a - \sqrt {{a^2} - b} }}{2}} , \hfill \\ {\text{если }}a \geqslant \sqrt b . \hfill \\ \end{array}\]

\[{\text{Вычислите }}\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt {2 + ...} } } } .\]

\[{\text{Докажите}}{\text{, что }}\sqrt 2 {\text{ - иррациональное число}}{\text{.}}\]

\[\begin{array}{l} {\text{Упростите:}} \hfill \\ \sqrt[5]{{\frac{{123 + 55\sqrt 5 }}{2}}} + \sqrt[5]{{\frac{{123 - 55\sqrt 5 }}{2}}}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Упростите:}} \hfill \\ \sqrt[6]{{99 + 70\sqrt 2 }} + \sqrt[6]{{99 - 70\sqrt 2 }}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Проверьте тождество:}} \hfill \\ \sqrt[3]{{\frac{1}{6}}} + \sqrt[3]{{\frac{4}{3}}} = \sqrt[3]{{\frac{9}{2}}}. \hfill \\ \end{array}\]

\[6 = \sqrt {1 + 5 \cdot \sqrt {1 + 6 \cdot \sqrt {1 + 7 \cdot \sqrt {1 + 8 \cdot \sqrt {1 + ....} } } } } \]

\[3 = \sqrt {1 + 2 \cdot \sqrt {1 + 3 \cdot \sqrt {1 + 4 \cdot \sqrt {1 + 5 \cdot \sqrt {1 + ...} } } } } \]

\[\begin{array}{l} {\text{Упростите:}} \hfill \\ \sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}}}} + \sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 - \sqrt 2 } \right)}^2}}}} . \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Упростите:}} \hfill \\ \sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 + \sqrt 2 } \right)}^4}}}} + \sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 - \sqrt 2 } \right)}^4}}}} . \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Упростите:}} \hfill \\ \sqrt[3]{{1 + {{\left( {\frac{1}{{\sqrt 3 + \sqrt 2 }}} \right)}^3}}} + \sqrt[3]{{1 + {{\left( {\frac{1}{{\sqrt 3 - \sqrt 2 }}} \right)}^3}}}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Упростите:}} \hfill \\ \sqrt[3]{{{{\left( {\sqrt 5 + 2} \right)}^{10}}}} + \sqrt[3]{{{{\left( {\sqrt 5 - 2} \right)}^{10}}}}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Проверьте тождество:}} \hfill \\ \sqrt[3]{{4 - 2\sqrt 2 }} + \sqrt[3]{{ - 1 + \sqrt 2 }} = \sqrt[3]{{3 + 2\sqrt 2 }}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Упростите:}} \hfill \\ \sqrt[4]{{\frac{4}{{2 + \sqrt[3]{4}}}}} + \sqrt[4]{{\frac{1}{{1 + \sqrt[3]{2}}}}}. \hfill \\ \end{array}\]

\[\sqrt[4]{{\frac{8}{{\sqrt[3]{2} + 2}}}} + \sqrt[4]{{\frac{{\sqrt[3]{2}}}{{\sqrt[3]{2} + 2}}}} = \sqrt[4]{{\frac{{4 + 7\sqrt[3]{2}}}{{2 - \sqrt[3]{2}}}}}\]