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受欢迎的代数 >

展开 (x^2-2/(x^3))^5

解答

展开

(x^{2} - \frac{2}{x ^{3}})^{5}

解答

x^{10} - 10 x^{5} + 40 - \frac{80}{x ^{5}} + \frac{80}{x ^{10}} - \frac{32}{x ^{15}}

求解步骤

(x^{2} - \frac{2}{x ^{3}})^{5}

使用二项式定理:

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

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

= i = 0 \sum 5 (i 5) (x^{2})^{(5 - i)} (- \frac{2}{x ^{3}})^{i}

展开求和

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

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} (x^{2})^{5} (- \frac{2}{x ^{3}})^{0} : x^{10}

化简

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

化简

\frac{5 !}{2 ! ( 5 - 2 ) !} (x^{2})^{3} (- \frac{2}{x ^{3}})^{2} : 40

化简

\frac{5 !}{3 ! ( 5 - 3 ) !} (x^{2})^{2} (- \frac{2}{x ^{3}})^{3} : - \frac{80}{x ^{5}}

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} (x^{2})^{1} (- \frac{2}{x ^{3}})^{4} : \frac{80}{x ^{10}}

化简

\frac{5 !}{5 ! ( 5 - 5 ) !} (x^{2})^{0} (- \frac{2}{x ^{3}})^{5} : - \frac{32}{x ^{15}}

= x^{10} - 10 x^{5} + 40 - \frac{80}{x ^{5}} + \frac{80}{x ^{10}} - \frac{32}{x ^{15}}

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