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受欢迎的代数 >

展开 (1/(3x)-2x^2)^6

解答

展开

(\frac{1}{3 x} - 2 x^{2})^{6}

解答

\frac{1}{729 x ^{6}} - \frac{4}{81 x ^{3}} + \frac{20}{27} - \frac{160 x ^{3}}{27} + \frac{80 x ^{6}}{3} - 64 x^{9} + 64 x^{12}

求解步骤

(\frac{1}{3 x} - 2 x^{2})^{6}

使用二项式定理:

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

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

= i = 0 \sum 6 (i 6) (\frac{1}{3 x})^{(6 - i)} (- 2 x^{2})^{i}

展开求和

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

化简

\frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{1}{3 x})^{6} (- 2 x^{2})^{0} : \frac{1}{729 x ^{6}}

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{1}{3 x})^{5} (- 2 x^{2})^{1} : - \frac{4}{81 x ^{3}}

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{1}{3 x})^{4} (- 2 x^{2})^{2} : \frac{20}{27}

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} (\frac{1}{3 x})^{3} (- 2 x^{2})^{3} : - \frac{160 x ^{3}}{27}

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{1}{3 x})^{2} (- 2 x^{2})^{4} : \frac{80 x ^{6}}{3}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} (\frac{1}{3 x})^{1} (- 2 x^{2})^{5} : - 64 x^{9}

化简

\frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{1}{3 x})^{0} (- 2 x^{2})^{6} : 64 x^{12}

= \frac{1}{729 x ^{6}} - \frac{4}{81 x ^{3}} + \frac{20}{27} - \frac{160 x ^{3}}{27} + \frac{80 x ^{6}}{3} - 64 x^{9} + 64 x^{12}

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e x p an d \frac{cos ^{2} ( x )}{( 1 - sin ( x ) ) ^{2}}

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