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展开 (5/4 x^2-1/(2x))^{10}

解答

展开

(\frac{5}{4} x^{2} - \frac{1}{2 x})^{10}

解答

\frac{9765625 x ^{20}}{1048576} - \frac{9765625 x ^{17}}{262144} + \frac{17578125 x ^{14}}{262144} - \frac{1171875 x ^{11}}{16384} + \frac{1640625 x ^{8}}{32768} - \frac{196875 x ^{5}}{8192} + \frac{65625 x ^{2}}{8192} - \frac{1875}{1024 x} + \frac{1125}{4096 x ^{4}} - \frac{25}{1024 x ^{7}} + \frac{1}{1024 x ^{10}}

求解步骤

(\frac{5}{4} x^{2} - \frac{1}{2 x})^{10}

使用二项式定理:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = \frac{5}{4} x^{2}, b = - \frac{1}{2 x}

= i = 0 \sum 10 (i 10) (\frac{5}{4} x^{2})^{(10 - i)} (- \frac{1}{2 x})^{i}

展开求和

= \frac{10 !}{0 ! ( 10 - 0 ) !} (\frac{5}{4} x^{2})^{10} (- \frac{1}{2 x})^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} (\frac{5}{4} x^{2})^{9} (- \frac{1}{2 x})^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} (\frac{5}{4} x^{2})^{8} (- \frac{1}{2 x})^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} (\frac{5}{4} x^{2})^{7} (- \frac{1}{2 x})^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} (\frac{5}{4} x^{2})^{6} (- \frac{1}{2 x})^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} (\frac{5}{4} x^{2})^{5} (- \frac{1}{2 x})^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} (\frac{5}{4} x^{2})^{4} (- \frac{1}{2 x})^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} (\frac{5}{4} x^{2})^{3} (- \frac{1}{2 x})^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} (\frac{5}{4} x^{2})^{2} (- \frac{1}{2 x})^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} (\frac{5}{4} x^{2})^{1} (- \frac{1}{2 x})^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} (\frac{5}{4} x^{2})^{0} (- \frac{1}{2 x})^{10}

化简

\frac{10 !}{0 ! ( 10 - 0 ) !} (\frac{5}{4} x^{2})^{10} (- \frac{1}{2 x})^{0} : \frac{9765625 x ^{20}}{1048576}

化简

\frac{10 !}{1 ! ( 10 - 1 ) !} (\frac{5}{4} x^{2})^{9} (- \frac{1}{2 x})^{1} : - \frac{9765625 x ^{17}}{262144}

化简

\frac{10 !}{2 ! ( 10 - 2 ) !} (\frac{5}{4} x^{2})^{8} (- \frac{1}{2 x})^{2} : \frac{17578125 x ^{14}}{262144}

化简

\frac{10 !}{3 ! ( 10 - 3 ) !} (\frac{5}{4} x^{2})^{7} (- \frac{1}{2 x})^{3} : - \frac{1171875 x ^{11}}{16384}

化简

\frac{10 !}{4 ! ( 10 - 4 ) !} (\frac{5}{4} x^{2})^{6} (- \frac{1}{2 x})^{4} : \frac{1640625 x ^{8}}{32768}

化简

\frac{10 !}{5 ! ( 10 - 5 ) !} (\frac{5}{4} x^{2})^{5} (- \frac{1}{2 x})^{5} : - \frac{196875 x ^{5}}{8192}

化简

\frac{10 !}{6 ! ( 10 - 6 ) !} (\frac{5}{4} x^{2})^{4} (- \frac{1}{2 x})^{6} : \frac{65625 x ^{2}}{8192}

化简

\frac{10 !}{7 ! ( 10 - 7 ) !} (\frac{5}{4} x^{2})^{3} (- \frac{1}{2 x})^{7} : - \frac{1875}{1024 x}

化简

\frac{10 !}{8 ! ( 10 - 8 ) !} (\frac{5}{4} x^{2})^{2} (- \frac{1}{2 x})^{8} : \frac{1125}{4096 x ^{4}}

化简

\frac{10 !}{9 ! ( 10 - 9 ) !} (\frac{5}{4} x^{2})^{1} (- \frac{1}{2 x})^{9} : - \frac{25}{1024 x ^{7}}

化简

\frac{10 !}{10 ! ( 10 - 10 ) !} (\frac{5}{4} x^{2})^{0} (- \frac{1}{2 x})^{10} : \frac{1}{1024 x ^{10}}

= \frac{9765625 x ^{20}}{1048576} - \frac{9765625 x ^{17}}{262144} + \frac{17578125 x ^{14}}{262144} - \frac{1171875 x ^{11}}{16384} + \frac{1640625 x ^{8}}{32768} - \frac{196875 x ^{5}}{8192} + \frac{65625 x ^{2}}{8192} - \frac{1875}{1024 x} + \frac{1125}{4096 x ^{4}} - \frac{25}{1024 x ^{7}} + \frac{1}{1024 x ^{10}}

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