Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 1114

\[\boxed{\mathbf{1114}\mathbf{.}}\]

\[1)\ \left( \sqrt[3]{x} + \frac{1}{\sqrt[3]{x}} \right)^{15}\]

\[C_{15}^{n} \bullet \left( x^{\frac{1}{3}} \right)^{15 - n} \bullet \left( x^{- \frac{1}{3}} \right)^{n} =\]

\[= C_{15}^{n} \bullet x^{5 - \frac{n}{3}} \bullet x^{- \frac{n}{3}} = C_{15}^{n} \bullet x^{5 - \frac{2n}{3}}\]

\[Номер\ члена,\ содержащего\ x^{\frac{1}{3}}:\]

\[x^{5 - \frac{2n}{3}} = x^{\frac{1}{3}}\]

\[5 - \frac{2n}{3} = \frac{1}{3}\]

\[15 - 2n = 1\]

\[2n = 14\]

\[n = 7.\]

\[Биноминальный\ \]

\[коэффициент:\]

\[C_{15}^{7} = \frac{15!}{(15 - 7)! \bullet 7!} = \frac{15!}{8! \bullet 7!} =\]

\[= \frac{15 \bullet 14 \bullet 13 \bullet 12 \bullet 11 \bullet 10 \bullet 9 \bullet 8!}{8! \bullet 7 \bullet 6 \bullet 5 \bullet 4 \bullet 3 \bullet 2} =\]

\[= 6\ 435.\]

\[Ответ:\ \ 6435x^{\frac{1}{3}}\text{.\ \ }\]

\[2)\ \left( \frac{1}{\sqrt{x}} + \sqrt{x} \right)^{14}\]

\[C_{14}^{n} \bullet \left( x^{- \frac{1}{2}} \right)^{14 - n} \bullet \left( x^{\frac{1}{2}} \right)^{n} =\]

\[= C_{14}^{n} \bullet x^{- 7 + \frac{n}{2}} \bullet x^{\frac{n}{2}} = C_{14}^{n} \bullet x^{n - 7}\]

\[Номер\ члена,\ содержащего\ x^{2}:\]

\[x^{n - 7} = x^{2}\]

\[n - 7 = 2\ \]

\[n = 9.\]

\[Биноминальный\ \]

\[коэффициент:\]

\[C_{14}^{9} = \frac{14!}{(14 - 9)! \bullet 9!} = \frac{14!}{5! \bullet 9!} =\]

\[= \frac{14 \bullet 13 \bullet 12 \bullet 11 \bullet 10 \bullet 9!}{5 \bullet 4 \bullet 3 \bullet 2 \bullet 9!} =\]

\[= 2002.\]

\[Ответ:\ \ 2002x^{2}\text{.\ }\]

\[3)\ \left( \frac{1}{\sqrt[4]{x}} + \sqrt[3]{x} \right)^{16}\]

\[C_{16}^{n} \bullet \left( x^{- \frac{1}{4}} \right)^{16 - n} \bullet \left( x^{\frac{1}{3}} \right)^{n} =\]

\[= C_{16}^{n} \bullet x^{- 4 + \frac{n}{4}} \bullet x^{\frac{n}{3}} = C_{16}^{n} \bullet x^{\frac{7n}{12} - 4}\]

\[Номер\ члена,\ содержащего\ \]

\[x^{- \frac{13}{12}}:\]

\[x^{\frac{7n}{12} - 4} = x^{- \frac{13}{12}}\]

\[\frac{n}{12} - 4 = - \frac{13}{12}\]

\[7n - 48 = - 13\]

\[7n = 3\ \]

\[n = 5.\]

\[Биноминальный\ \]

\[коэффициент:\]

\[C_{16}^{5} = \frac{16!}{(16 - 5)! \bullet 5!} = \frac{16!}{11! \bullet 5!} =\]

\[= \frac{16 \bullet 15 \bullet 14 \bullet 13 \bullet 12 \bullet 11!}{11! \bullet 5 \bullet 4 \bullet 3 \bullet 2} =\]

\[= 4368.\]

\[Ответ:\ \ 4368x^{- \frac{13}{12}}\text{.\ \ }\]

\[4)\ \left( \sqrt[5]{x} + \frac{1}{\sqrt[3]{x}} \right)^{13}\]

\[C_{13}^{n} \bullet \left( x^{\frac{1}{5}} \right)^{13 - n} \bullet \left( x^{- \frac{1}{3}} \right)^{n} =\]

\[= C_{13}^{n} \bullet x^{\frac{13}{5} - \frac{n}{5}} \bullet x^{- \frac{n}{3}} =\]

\[= C_{13}^{n} \bullet x^{\frac{13}{5} - \frac{8n}{15}}\]

\[Номер\ члена,\ содержащего\ \]

\[x^{- 0,6}:\]

\[x^{\frac{13}{5} - \frac{8n}{15}} = x^{- 0,6}\]

\[\frac{13}{5} - \frac{8n}{15} = - 0,6\]

\[39 - 8n = - 9\]

\[8n = 48\ \]

\[n = 6.\]

\[Биноминальный\ \]

\[коэффициент:\]

\[C_{13}^{6} = \frac{13!}{(13 - 6)! \bullet 6!} = \frac{13!}{7! \bullet 6!} =\]

\[= \frac{13 \bullet 12 \bullet 11 \bullet 10 \bullet 9 \bullet 8 \bullet 7!}{7! \bullet 6 \bullet 5 \bullet 4 \bullet 3 \bullet 2} =\]

\[= 1716.\]

\[Ответ:\ \ 1716x^{- 0,6}.\]