tag:
radicals
§
Формулы сложных радикалов
\[\sqrt {a \pm \sqrt b } = \sqrt {\frac{{a + \sqrt {{a^2} - b} }}{2}} \pm \sqrt {\frac{{a - \sqrt {{a^2} - b} }}{2}} \]
§
\[\begin{array}{l}
{\text{Пусть }}a,b,c{\text{ - действительные корни многочлена }}{x^3} - p{x^2} + qx - r{\text{,}} \hfill \\
{\text{причём }}3\sqrt[3]{{{r^2}}} + p\sqrt[3]{r} + q = 0. \hfill \\
{\text{Тогда }}\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = \sqrt[3]{{3\sqrt[3]{{{p^2}\sqrt[3]{r} - 3q\sqrt[3]{r}}} - p - 6\sqrt[3]{r}}}. \hfill \\
\end{array}\]
526.
$$\eqalign{
{\text{Докажите}}{\text{, что}} \hfill \\
\cos \frac{\pi }{{{2^n}}} = \frac{1}{2}\sqrt {2 + \sqrt {2 + \sqrt {2 + ... + \sqrt 2 } } } \hfill \\
(n - 1{\text{ знак корня}}{\text{, }}n \in \mathbb{N}) \hfill \\
} $$
1011.
\[\begin{array}{l}
{\text{Докажите равенство:}} \hfill \\
\sqrt {a \pm \sqrt b } = \sqrt {\frac{{a + \sqrt {{a^2} - b} }}{2}} \pm \sqrt {\frac{{a - \sqrt {{a^2} - b} }}{2}} , \hfill \\
{\text{если }}a \geqslant \sqrt b . \hfill \\
\end{array}\]
1410.
\[\begin{array}{l}
\sqrt[3]{{\cos \frac{{2\pi }}{7}}} + \sqrt[3]{{\cos \frac{{4\pi }}{7}}} + \sqrt[3]{{\cos \frac{{8\pi }}{7}}} = \sqrt[3]{{\frac{{5 - 3\sqrt[3]{7}}}{2}}} \hfill \\
\sqrt[3]{{\cos \frac{{2\pi }}{9}}} + \sqrt[3]{{\cos \frac{{4\pi }}{9}}} + \sqrt[3]{{\cos \frac{{8\pi }}{9}}} = \sqrt[3]{{\frac{{3\sqrt[3]{9} - 6}}{2}}} \hfill \\
\end{array}\]
1605.
\[\begin{array}{l}
{\text{Докажите}}{\text{, что при всех натуральных }}n{\text{ верно неравенство:}} \hfill \\
\sqrt {{1^3} + \sqrt {{2^3} + ... + \sqrt {{n^3}} } } < 3. \hfill \\
\end{array}\]
1875.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[5]{{232 + 164\sqrt 2 }} + \sqrt[5]{{232 - 164\sqrt 2 }}. \hfill \\
\end{array}\]
1876.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[3]{{\frac{{27 + 11\sqrt 6 }}{{72}}}} + \sqrt[3]{{\frac{{27 - 11\sqrt 6 }}{{72}}}}. \hfill \\
\end{array}\]
1883.
\[\sqrt {1 + \sqrt {\frac{1}{2} + \sqrt {\frac{1}{3} + \sqrt {\frac{1}{4} + ... + \sqrt {\frac{1}{n} + ...} } } } } = ?\]
1884.
\[\begin{array}{l}
{\text{Решите уравнение:}} \hfill \\
\sqrt[4]{{\frac{{3 + 2x}}{{3 - 2x}}}} + \sqrt[4]{{\frac{{3 - 2x}}{{3 + 2x}}}} = {x^2} + 3. \hfill \\
\end{array}\]
1886.
\[\begin{array}{l}
{\text{а) }}\sqrt {\sqrt[3]{{80}} - \sqrt[3]{{64}}} = \frac{1}{3} \cdot \left( { - \sqrt[3]{{100}} + 2\sqrt[3]{{10}} + 2} \right) \hfill \\
{\text{б) }}\sqrt {\sqrt[3]{{125}} - \sqrt[3]{{100}}} = \frac{1}{3} \cdot \left( {\sqrt[3]{{100}} + \sqrt[3]{{10}} - 5} \right) \hfill \\
\end{array}\]
1888.
\[\begin{array}{l}
\left[ {{\text{Рамануджан}}} \right] \hfill \\
\sqrt[3]{{\sqrt[3]{2} - 1}} = \sqrt[3]{{\frac{1}{9}}} - \sqrt[3]{{\frac{2}{9}}} + \sqrt[3]{{\frac{4}{9}}} \hfill \\
\end{array}\]
1890.
\[\begin{array}{l}
\left[ {{\text{Рамануджан}}} \right] \hfill \\
\sqrt[4]{{\frac{{3 + 2\sqrt[4]{5}}}{{3 - 2\sqrt[4]{5}}}}} = \frac{{\sqrt[4]{5} + 1}}{{\sqrt[4]{5} - 1}} \hfill \\
\end{array}\]
1891.
\[\begin{array}{l}
{\text{Проверьте верность равенства:}} \hfill \\
\sqrt[4]{{\frac{{2 + \sqrt[4]{{12}}}}{{2 - \sqrt[4]{{12}}}}}} = \frac{{3 + \sqrt 3 + \sqrt[4]{{12}}}}{{3 + \sqrt 3 - \sqrt[4]{{12}}}}. \hfill \\
\end{array}\]
1893.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[4]{{\frac{{3 + 2\sqrt[4]{2}}}{{3 - 2\sqrt[4]{2}}}}} + \sqrt[4]{{\frac{{3 - 2\sqrt[4]{2}}}{{3 + 2\sqrt[4]{2}}}}}. \hfill \\
\end{array}\]
1896.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt[4]{{\frac{{11 + 6\sqrt[4]{{10}}}}{{11 - 6\sqrt[4]{{10}}}}}} = \frac{1}{{41}}\left( {\sqrt {1189 + 451\sqrt {10} } + \sqrt {2870 + 451\sqrt {10} } } \right). \hfill \\
\end{array}\]
1898.
\[\begin{array}{l}
{\text{Докажите тождество:}} \hfill \\
\sqrt[8]{{\frac{{3 + 2\sqrt[4]{2}}}{{3 - 2\sqrt[4]{2}}}}} + \sqrt[8]{{\frac{{3 - 2\sqrt[4]{2}}}{{3 + 2\sqrt[4]{2}}}}} = \frac{1}{7}\sqrt {98 + 14\sqrt {35 + 21\sqrt 2 } } . \hfill \\
\end{array}\]
1899.
\[\begin{array}{l}
{\text{Докажите тождество:}} \hfill \\
\sqrt[8]{{\frac{{3 + 2\sqrt[4]{2}}}{{3 - 2\sqrt[4]{2}}}}} = \frac{1}{{14}}\left( {\sqrt {98 + 14\sqrt {35 + 21\sqrt 2 } } + \sqrt { - 98 + 14\sqrt {35 + 21\sqrt 2 } } } \right). \hfill \\
\end{array}\]
1900.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[5]{{\frac{{123 + 55\sqrt 5 }}{2}}} + \sqrt[5]{{\frac{{123 - 55\sqrt 5 }}{2}}}. \hfill \\
\end{array}\]
1901.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[6]{{99 + 70\sqrt 2 }} + \sqrt[6]{{99 - 70\sqrt 2 }}. \hfill \\
\end{array}\]
1902.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt {4 + \sqrt[3]{2} + 2\sqrt[3]{4}} + \sqrt { - 4 + \sqrt[3]{2} + 2\sqrt[3]{4}} = \sqrt[6]{{2048}}. \hfill \\
\end{array}\]
1904.
\[\begin{array}{l}
{\text{Докажите тождество:}} \hfill \\
\sqrt[{{2^n}}]{{\frac{{2 + \sqrt 2 }}{{2 - \sqrt 2 }}}} + \sqrt[{{2^n}}]{{\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}}} = \underbrace {\sqrt {2 + \sqrt {2 + \sqrt {2 + ... + \sqrt {2 + 2\sqrt 2 } } } } }_{n{\text{ корней}}}. \hfill \\
\end{array}\]
1907.
\[\begin{array}{l}
{\text{Пусть }}a,b,c{\text{ - положительные рациональные числа}}{\text{,}} \hfill \\
{\text{а числа }}\sqrt[n]{a}{\text{, }}\sqrt[n]{b},{\text{ }}\sqrt[n]{c}{\text{ - иррациональные}}{\text{, }}n \in \mathbb{N}. \hfill \\
{\text{Возможны ли при каком - либо }}n \geqslant 3{\text{ равенства:}} \hfill \\
{\text{а) }}\sqrt[n]{a} + \sqrt[n]{b} = 1; \hfill \\
{\text{б) }}\sqrt[n]{a} + \sqrt[n]{b} = \sqrt[n]{c} + 1? \hfill \\
\end{array}\]
1912.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt {\frac{{3 + \sqrt[3]{3}}}{{3 - \sqrt[3]{3}}}} + \sqrt {\frac{{3 - \sqrt[3]{3}}}{{3 + \sqrt[3]{3}}}} . \hfill \\
\end{array}\]
1922.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt {\frac{{\sqrt[3]{3} - \sqrt[3]{2}}}{{\sqrt[3]{3} + \sqrt[3]{2}}}} + \sqrt {\frac{{\sqrt[3]{3} + \sqrt[3]{2}}}{{\sqrt[3]{3} - \sqrt[3]{2}}}} . \hfill \\
\end{array}\]
1923.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt {\sqrt {\frac{1}{2}} + \sqrt {\frac{1}{3}} } + \sqrt {\sqrt {\frac{1}{2}} - \sqrt {\frac{1}{3}} } = \sqrt[4]{{\frac{{8 + 4\sqrt 3 }}{3}}}. \hfill \\
\end{array}\]
1924.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt[4]{{\sqrt {\frac{1}{6}} + \sqrt {\frac{1}{8}} }} + \sqrt[4]{{\sqrt {\frac{1}{6}} - \sqrt {\frac{1}{8}} }} = \sqrt[4]{{2 + \frac{{5\sqrt 6 }}{6}}}. \hfill \\
\end{array}\]
1925.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt[4]{{\sqrt[4]{{\frac{1}{6}}} + \sqrt[4]{{\frac{1}{8}}}}} + \sqrt[4]{{\sqrt[4]{{\frac{1}{6}}} - \sqrt[4]{{\frac{1}{8}}}}} = \sqrt[{16}]{{378 + 216\sqrt 3 }}. \hfill \\
\end{array}\]
1926.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt[4]{{\sqrt[4]{{\frac{1}{9}}} + \sqrt[4]{{\frac{1}{{12}}}}}} + \sqrt[4]{{\sqrt[4]{{\frac{1}{9}}} - \sqrt[4]{{\frac{1}{{12}}}}}} = \sqrt[4]{{3 + \sqrt 3 }}. \hfill \\
\end{array}\]
1928.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt[3]{{\frac{1}{6}}} + \sqrt[3]{{\frac{4}{3}}} = \sqrt[3]{{\frac{9}{2}}}. \hfill \\
\end{array}\]
1929.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\sqrt[4]{{\sqrt[4]{4} + \sqrt[4]{3}}} = \frac{1}{2}\left( {\sqrt[8]{{72 + 36\sqrt 3 }} + \sqrt[8]{{104 - 60\sqrt 3 }}} \right). \hfill \\
\end{array}\]
1932.
\[{\text{Докажите}}{\text{, что }}\sqrt 2 ,\sqrt[3]{2},\sqrt[4]{2},...{\text{ линейно независимы над }}\mathbb{Q}.\]
1933.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\frac{{\sqrt[3]{{378}} + \sqrt[3]{{441}}}}{{\sqrt[3]{{28}} + \sqrt[3]{{24}}}}. \hfill \\
\end{array}\]
1934.
\[\begin{array}{l}
{\text{Проверьте тождество:}} \hfill \\
\frac{{\sqrt[3]{{14}} - \sqrt[3]{7}}}{{\sqrt[3]{{16}} - \sqrt[3]{4}}} = \sqrt[3]{{\frac{7}{{12}}\left( {\sqrt[3]{2} - 1} \right)}}. \hfill \\
\end{array}\]
1939.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}}}} + \sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 - \sqrt 2 } \right)}^2}}}} . \hfill \\
\end{array}\]
1940.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 + \sqrt 2 } \right)}^4}}}} + \sqrt {1 + \frac{1}{{{{\left( {\sqrt 3 - \sqrt 2 } \right)}^4}}}} . \hfill \\
\end{array}\]
1941.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[3]{{1 + {{\left( {\frac{1}{{\sqrt 3 + \sqrt 2 }}} \right)}^3}}} + \sqrt[3]{{1 + {{\left( {\frac{1}{{\sqrt 3 - \sqrt 2 }}} \right)}^3}}}. \hfill \\
\end{array}\]
1942.
\[\begin{array}{l}
{\text{Пусть }}a,b,c{\text{ - корни многочлена }}{x^3} - {x^2} - 2x + 1. \hfill \\
{\text{Докажите тождества:}} \hfill \\
{\text{а) }}\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = \sqrt[3]{{3\sqrt[3]{7} - 5}}; \hfill \\
{\text{б) }}\sqrt[3]{{a + b}} + \sqrt[3]{{a + c}} + \sqrt[3]{{b + c}} = \sqrt[3]{{3\sqrt[3]{7} - 4}}; \hfill \\
{\text{в) }}\sqrt[3]{{a{b^2}}} + \sqrt[3]{{{a^2}c}} + \sqrt[3]{{b{c^2}}} = \sqrt[3]{7}; \hfill \\
{\text{г) }}\sqrt[3]{{\frac{{{a^2}}}{{{b^2}}}}} + \sqrt[3]{{\frac{{{b^2}}}{{{c^2}}}}} + \sqrt[3]{{\frac{{{c^2}}}{{{a^2}}}}} = \sqrt[3]{{49}}; \hfill \\
{\text{д) }}\sqrt[3]{{\frac{{{a^4}}}{{{b^2}{c^2}}}}} + \sqrt[3]{{\frac{{{b^4}}}{{{a^2}{c^2}}}}} + \sqrt[3]{{\frac{{{c^4}}}{{{a^2}{b^2}}}}} = 5; \hfill \\
{\text{е) }}\frac{1}{{\sqrt[3]{{a + 1}}}} + \frac{1}{{\sqrt[3]{{b + 1}}}} + \frac{1}{{\sqrt[3]{{c + 1}}}} = 0. \hfill \\
\end{array}\]
1945.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[4]{{\frac{4}{{2 + \sqrt[3]{4}}}}} + \sqrt[4]{{\frac{1}{{1 + \sqrt[3]{2}}}}}. \hfill \\
\end{array}\]
1946.
\[\sqrt[4]{{\frac{8}{{\sqrt[3]{2} + 2}}}} + \sqrt[4]{{\frac{{\sqrt[3]{2}}}{{\sqrt[3]{2} + 2}}}} = \sqrt[4]{{\frac{{4 + 7\sqrt[3]{2}}}{{2 - \sqrt[3]{2}}}}}\]
1947.
\[\sqrt[4]{{\frac{{24}}{{\sqrt[3]{3} + 5}} - 3}} + \frac{2}{3}\sqrt {1 - \frac{6}{{\sqrt[3]{3} + 5}}} = \sqrt {\frac{2}{{5 \cdot \left( {5\sqrt[3]{3} - 7} \right)}} - \frac{{31}}{{45}}} \]
1948.
\[\begin{array}{l}
{\text{а) }}\sqrt {x + \frac{1}{2}\sqrt {x + \frac{1}{4}\sqrt {x + \frac{1}{8}\sqrt {x + \frac{1}{{16}}\sqrt {x + ...} } } } } = \frac{1}{4} + \sqrt x ; \hfill \\
{\text{б) }}\sqrt {x - \frac{1}{2}\sqrt {x - \frac{1}{4}\sqrt {x - \frac{1}{8}\sqrt {x - \frac{1}{{16}}\sqrt {x - ...} } } } } = - \frac{1}{4} + \sqrt x . \hfill \\
\end{array}\]
2051.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
\sqrt[4]{{\frac{{\sqrt 6 }}{2} - \frac{{\sqrt 2 }}{2}}} + \sqrt[4]{{\frac{{\sqrt 6 }}{2} + \frac{{\sqrt 2 }}{2}}}. \hfill \\
\end{array}\]
2053.
\[\begin{array}{l}
{\text{Упростите:}} \hfill \\
{\left( {\frac{1}{{\sqrt[4]{5} + \sqrt[4]{3}}}} \right)^4} + {\left( {\frac{1}{{\sqrt[4]{5} - \sqrt[4]{3}}}} \right)^4}. \hfill \\
\end{array}\]
2100.
\[\begin{array}{l}
{\text{Simplify}}: \hfill \\
\sqrt {\sqrt 6 + \sqrt 2 + 1} + \sqrt {\sqrt 6 + \sqrt 2 - 1} + \sqrt {\sqrt 6 - \sqrt 2 + 1} + \sqrt {\sqrt 6 - \sqrt 2 - 1} . \hfill \\
\end{array} \]
2124.
$$\begin{array}{l}
{\text{Найдите целое число }}x,{\text{ которое удовлетворяет равенству}}{\text{.}} \hfill \\
\sqrt[4]{{3\sqrt 5 - 1 - \sqrt {6\left( {5 - \sqrt 5 } \right)} }} + \sqrt[4]{{3\sqrt 5 - 1 + \sqrt {6\left( {5 - \sqrt 5 } \right)} }} = \sqrt {x + \sqrt {6 + 6\sqrt 5 } } . \hfill \\
\end{array} $$
2126.
\[{\text{Найдите }}x \in \mathbb{N}.\]
$$\frac{1}{{\sqrt[3]{{\cos \frac{{2\pi }}{9}}}}} + \frac{1}{{\sqrt[3]{{\cos \frac{{4\pi }}{9}}}}} + \frac{1}{{\sqrt[3]{{\cos \frac{{8\pi }}{9}}}}} = \sqrt[3]{{x\left( {\sqrt[3]{9} - 1} \right)}}$$
2127.
\[{\text{Найдите }}y \in \mathbb{N}.\]
$$\frac{1}{{\sqrt[3]{{{{\cos }^2}\frac{{2\pi }}{9}}}}} + \frac{1}{{\sqrt[3]{{{{\cos }^2}\frac{{4\pi }}{9}}}}} + \frac{1}{{\sqrt[3]{{{{\cos }^2}\frac{{8\pi }}{9}}}}} = \sqrt[3]{{\frac{{y\left( {\sqrt[3]{9} - 1} \right)}}{{\sqrt[3]{9} - 2}}}}$$