\[\begin{array}{l} {\text{а) }}\sqrt {\sqrt[3]{{80}} - \sqrt[3]{{64}}} = \frac{1}{3} \cdot \left( { - \sqrt[3]{{100}} + 2\sqrt[3]{{10}} + 2} \right) \hfill \\ {\text{б) }}\sqrt {\sqrt[3]{{125}} - \sqrt[3]{{100}}} = \frac{1}{3} \cdot \left( {\sqrt[3]{{100}} + \sqrt[3]{{10}} - 5} \right) \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Проверьте верность равенства:}} \hfill \\ \sqrt[4]{{\frac{{2 + \sqrt[4]{{12}}}}{{2 - \sqrt[4]{{12}}}}}} = \frac{{3 + \sqrt 3 + \sqrt[4]{{12}}}}{{3 + \sqrt 3 - \sqrt[4]{{12}}}}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Проверьте тождество:}} \hfill \\ \sqrt {4 + \sqrt[3]{2} + 2\sqrt[3]{4}} + \sqrt { - 4 + \sqrt[3]{2} + 2\sqrt[3]{4}} = \sqrt[6]{{2048}}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Докажите тождество:}} \hfill \\ \sqrt[8]{{\frac{{3 + 2\sqrt[4]{2}}}{{3 - 2\sqrt[4]{2}}}}} + \sqrt[8]{{\frac{{3 - 2\sqrt[4]{2}}}{{3 + 2\sqrt[4]{2}}}}} = \frac{1}{7}\sqrt {98 + 14\sqrt {35 + 21\sqrt 2 } } . \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Докажите тождество:}} \hfill \\ \sqrt[8]{{\frac{{3 + 2\sqrt[4]{2}}}{{3 - 2\sqrt[4]{2}}}}} = \frac{1}{{14}}\left( {\sqrt {98 + 14\sqrt {35 + 21\sqrt 2 } } + \sqrt { - 98 + 14\sqrt {35 + 21\sqrt 2 } } } \right). \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Докажите тождество:}} \hfill \\ \sqrt[{{2^n}}]{{\frac{{2 + \sqrt 2 }}{{2 - \sqrt 2 }}}} + \sqrt[{{2^n}}]{{\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}}} = \underbrace {\sqrt {2 + \sqrt {2 + \sqrt {2 + ... + \sqrt {2 + 2\sqrt 2 } } } } }_{n{\text{ корней}}}. \hfill \\ \end{array}\]

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

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

\[\begin{array}{l} {\text{Проверьте тождество:}} \hfill \\ \sqrt {\sqrt {\frac{1}{2}} + \sqrt {\frac{1}{3}} } + \sqrt {\sqrt {\frac{1}{2}} - \sqrt {\frac{1}{3}} } = \sqrt[4]{{\frac{{8 + 4\sqrt 3 }}{3}}}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Проверьте тождество:}} \hfill \\ \sqrt[4]{{\sqrt {\frac{1}{6}} + \sqrt {\frac{1}{8}} }} + \sqrt[4]{{\sqrt {\frac{1}{6}} - \sqrt {\frac{1}{8}} }} = \sqrt[4]{{2 + \frac{{5\sqrt 6 }}{6}}}. \hfill \\ \end{array}\]

\[\begin{array}{l} {\text{Проверьте тождество:}} \hfill \\ \frac{{\sqrt[3]{{14}} - \sqrt[3]{7}}}{{\sqrt[3]{{16}} - \sqrt[3]{4}}} = \sqrt[3]{{\frac{7}{{12}}\left( {\sqrt[3]{2} - 1} \right)}}. \hfill \\ \end{array}\]