Корни многочленов
1942.
\[\begin{array}{l}
{\text{Пусть }}a,b,c{\text{ - корни многочлена }}{x^3} - {x^2} - 2x + 1. \hfill \\
{\text{Докажите тождества:}} \hfill \\
{\text{а) }}\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = \sqrt[3]{{3\sqrt[3]{7} - 5}}; \hfill \\
{\text{б) }}\sqrt[3]{{a + b}} + \sqrt[3]{{a + c}} + \sqrt[3]{{b + c}} = \sqrt[3]{{3\sqrt[3]{7} - 4}}; \hfill \\
{\text{в) }}\sqrt[3]{{a{b^2}}} + \sqrt[3]{{{a^2}c}} + \sqrt[3]{{b{c^2}}} = \sqrt[3]{7}; \hfill \\
{\text{г) }}\sqrt[3]{{\frac{{{a^2}}}{{{b^2}}}}} + \sqrt[3]{{\frac{{{b^2}}}{{{c^2}}}}} + \sqrt[3]{{\frac{{{c^2}}}{{{a^2}}}}} = \sqrt[3]{{49}}; \hfill \\
{\text{д) }}\sqrt[3]{{\frac{{{a^4}}}{{{b^2}{c^2}}}}} + \sqrt[3]{{\frac{{{b^4}}}{{{a^2}{c^2}}}}} + \sqrt[3]{{\frac{{{c^4}}}{{{a^2}{b^2}}}}} = 5; \hfill \\
{\text{е) }}\frac{1}{{\sqrt[3]{{a + 1}}}} + \frac{1}{{\sqrt[3]{{b + 1}}}} + \frac{1}{{\sqrt[3]{{c + 1}}}} = 0. \hfill \\
\end{array}\]
1410.
\[\begin{array}{l}
\sqrt[3]{{\cos \frac{{2\pi }}{7}}} + \sqrt[3]{{\cos \frac{{4\pi }}{7}}} + \sqrt[3]{{\cos \frac{{8\pi }}{7}}} = \sqrt[3]{{\frac{{5 - 3\sqrt[3]{7}}}{2}}} \hfill \\
\sqrt[3]{{\cos \frac{{2\pi }}{9}}} + \sqrt[3]{{\cos \frac{{4\pi }}{9}}} + \sqrt[3]{{\cos \frac{{8\pi }}{9}}} = \sqrt[3]{{\frac{{3\sqrt[3]{9} - 6}}{2}}} \hfill \\
\end{array}\]