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受欢迎的代数 >

展开 (x^3-1/(x^6))^6

解答

展开

(x^{3} - \frac{1}{x ^{6}})^{6}

解答

x^{18} - 6 x^{9} + 15 - \frac{20}{x ^{9}} + \frac{15}{x ^{18}} - \frac{6}{x ^{27}} + \frac{1}{x ^{36}}

求解步骤

(x^{3} - \frac{1}{x ^{6}})^{6}

使用二项式定理:

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

a = x^{3}, b = - \frac{1}{x ^{6}}

= i = 0 \sum 6 (i 6) (x^{3})^{(6 - i)} (- \frac{1}{x ^{6}})^{i}

展开求和

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

化简

\frac{6 !}{0 ! ( 6 - 0 ) !} (x^{3})^{6} (- \frac{1}{x ^{6}})^{0} : x^{18}

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} (x^{3})^{5} (- \frac{1}{x ^{6}})^{1} : - 6 x^{9}

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} (x^{3})^{4} (- \frac{1}{x ^{6}})^{2} : 15

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} (x^{3})^{3} (- \frac{1}{x ^{6}})^{3} : - \frac{20}{x ^{9}}

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} (x^{3})^{2} (- \frac{1}{x ^{6}})^{4} : \frac{15}{x ^{18}}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} (x^{3})^{1} (- \frac{1}{x ^{6}})^{5} : - \frac{6}{x ^{27}}

化简

\frac{6 !}{6 ! ( 6 - 6 ) !} (x^{3})^{0} (- \frac{1}{x ^{6}})^{6} : \frac{1}{x ^{36}}

= x^{18} - 6 x^{9} + 15 - \frac{20}{x ^{9}} + \frac{15}{x ^{18}} - \frac{6}{x ^{27}} + \frac{1}{x ^{36}}

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