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受欢迎的代数 >

展开 (10x^3-1/(2x^2))^5

解答

展开

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

解答

100000 x^{15} - 25000 x^{10} + 2500 x^{5} - 125 + \frac{25}{8 x ^{5}} - \frac{1}{32 x ^{10}}

求解步骤

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

使用二项式定理:

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

a = 10 x^{3}, b = - \frac{1}{2 x ^{2}}

= i = 0 \sum 5 (i 5) (10 x^{3})^{(5 - i)} (- \frac{1}{2 x ^{2}})^{i}

展开求和

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

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} (10 x^{3})^{5} (- \frac{1}{2 x ^{2}})^{0} : 100000 x^{15}

化简

\frac{5 !}{1 ! ( 5 - 1 ) !} (10 x^{3})^{4} (- \frac{1}{2 x ^{2}})^{1} : - 25000 x^{10}

化简

\frac{5 !}{2 ! ( 5 - 2 ) !} (10 x^{3})^{3} (- \frac{1}{2 x ^{2}})^{2} : 2500 x^{5}

化简

\frac{5 !}{3 ! ( 5 - 3 ) !} (10 x^{3})^{2} (- \frac{1}{2 x ^{2}})^{3} : - 125

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} (10 x^{3})^{1} (- \frac{1}{2 x ^{2}})^{4} : \frac{25}{8 x ^{5}}

化简

\frac{5 !}{5 ! ( 5 - 5 ) !} (10 x^{3})^{0} (- \frac{1}{2 x ^{2}})^{5} : - \frac{1}{32 x ^{10}}

= 100000 x^{15} - 25000 x^{10} + 2500 x^{5} - 125 + \frac{25}{8 x ^{5}} - \frac{1}{32 x ^{10}}

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