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受欢迎的代数 >

展开 (3x-4/x)^8

解答

展开

(3 x - \frac{4}{x})^{8}

解答

6561 x^{8} - 69984 x^{6} + 326592 x^{4} - 870912 x^{2} + 1451520 - \frac{1548288}{x ^{2}} + \frac{1032192}{x ^{4}} - \frac{393216}{x ^{6}} + \frac{65536}{x ^{8}}

求解步骤

(3 x - \frac{4}{x})^{8}

使用二项式定理:

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

a = 3 x, b = - \frac{4}{x}

= i = 0 \sum 8 (i 8) (3 x)^{(8 - i)} (- \frac{4}{x})^{i}

展开求和

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

化简

\frac{8 !}{0 ! ( 8 - 0 ) !} (3 x)^{8} (- \frac{4}{x})^{0} : 6561 x^{8}

化简

\frac{8 !}{1 ! ( 8 - 1 ) !} (3 x)^{7} (- \frac{4}{x})^{1} : - 69984 x^{6}

化简

\frac{8 !}{2 ! ( 8 - 2 ) !} (3 x)^{6} (- \frac{4}{x})^{2} : 326592 x^{4}

化简

\frac{8 !}{3 ! ( 8 - 3 ) !} (3 x)^{5} (- \frac{4}{x})^{3} : - 870912 x^{2}

化简

\frac{8 !}{4 ! ( 8 - 4 ) !} (3 x)^{4} (- \frac{4}{x})^{4} : 1451520

化简

\frac{8 !}{5 ! ( 8 - 5 ) !} (3 x)^{3} (- \frac{4}{x})^{5} : - \frac{1548288}{x ^{2}}

化简

\frac{8 !}{6 ! ( 8 - 6 ) !} (3 x)^{2} (- \frac{4}{x})^{6} : \frac{1032192}{x ^{4}}

化简

\frac{8 !}{7 ! ( 8 - 7 ) !} (3 x)^{1} (- \frac{4}{x})^{7} : - \frac{393216}{x ^{6}}

化简

\frac{8 !}{8 ! ( 8 - 8 ) !} (3 x)^{0} (- \frac{4}{x})^{8} : \frac{65536}{x ^{8}}

= 6561 x^{8} - 69984 x^{6} + 326592 x^{4} - 870912 x^{2} + 1451520 - \frac{1548288}{x ^{2}} + \frac{1032192}{x ^{4}} - \frac{393216}{x ^{6}} + \frac{65536}{x ^{8}}

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