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受欢迎的代数 >

展开 (1/(2x^2)-x^3)^{10}

解答

展开

(\frac{1}{2 x ^{2}} - x^{3})^{10}

解答

\frac{1}{1024 x ^{20}} - \frac{5}{256 x ^{15}} + \frac{45}{256 x ^{10}} - \frac{15}{16 x ^{5}} + \frac{105}{32} - \frac{63 x ^{5}}{8} + \frac{105 x ^{10}}{8} - 15 x^{15} + \frac{45 x ^{20}}{4} - 5 x^{25} + x^{30}

求解步骤

(\frac{1}{2 x ^{2}} - x^{3})^{10}

使用二项式定理:

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

a = \frac{1}{2 x ^{2}}, b = - x^{3}

= i = 0 \sum 10 (i 10) (\frac{1}{2 x ^{2}})^{(10 - i)} (- x^{3})^{i}

展开求和

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

化简

\frac{10 !}{0 ! ( 10 - 0 ) !} (\frac{1}{2 x ^{2}})^{10} (- x^{3})^{0} : \frac{1}{1024 x ^{20}}

化简

\frac{10 !}{1 ! ( 10 - 1 ) !} (\frac{1}{2 x ^{2}})^{9} (- x^{3})^{1} : - \frac{5}{256 x ^{15}}

化简

\frac{10 !}{2 ! ( 10 - 2 ) !} (\frac{1}{2 x ^{2}})^{8} (- x^{3})^{2} : \frac{45}{256 x ^{10}}

化简

\frac{10 !}{3 ! ( 10 - 3 ) !} (\frac{1}{2 x ^{2}})^{7} (- x^{3})^{3} : - \frac{15}{16 x ^{5}}

化简

\frac{10 !}{4 ! ( 10 - 4 ) !} (\frac{1}{2 x ^{2}})^{6} (- x^{3})^{4} : \frac{105}{32}

化简

\frac{10 !}{5 ! ( 10 - 5 ) !} (\frac{1}{2 x ^{2}})^{5} (- x^{3})^{5} : - \frac{63 x ^{5}}{8}

化简

\frac{10 !}{6 ! ( 10 - 6 ) !} (\frac{1}{2 x ^{2}})^{4} (- x^{3})^{6} : \frac{105 x ^{10}}{8}

化简

\frac{10 !}{7 ! ( 10 - 7 ) !} (\frac{1}{2 x ^{2}})^{3} (- x^{3})^{7} : - 15 x^{15}

化简

\frac{10 !}{8 ! ( 10 - 8 ) !} (\frac{1}{2 x ^{2}})^{2} (- x^{3})^{8} : \frac{45 x ^{20}}{4}

化简

\frac{10 !}{9 ! ( 10 - 9 ) !} (\frac{1}{2 x ^{2}})^{1} (- x^{3})^{9} : - 5 x^{25}

化简

\frac{10 !}{10 ! ( 10 - 10 ) !} (\frac{1}{2 x ^{2}})^{0} (- x^{3})^{10} : x^{30}

= \frac{1}{1024 x ^{20}} - \frac{5}{256 x ^{15}} + \frac{45}{256 x ^{10}} - \frac{15}{16 x ^{5}} + \frac{105}{32} - \frac{63 x ^{5}}{8} + \frac{105 x ^{10}}{8} - 15 x^{15} + \frac{45 x ^{20}}{4} - 5 x^{25} + x^{30}

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