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受欢迎的代数 >

展开 (x^2+1/x)^{12}

解答

展开

(x^{2} + \frac{1}{x})^{12}

解答

x^{24} + 12 x^{21} + 66 x^{18} + 220 x^{15} + 495 x^{12} + 792 x^{9} + 924 x^{6} + 792 x^{3} + 495 + \frac{220}{x ^{3}} + \frac{66}{x ^{6}} + \frac{12}{x ^{9}} + \frac{1}{x ^{12}}

求解步骤

(x^{2} + \frac{1}{x})^{12}

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{1}{x}

= i = 0 \sum 12 (i 12) (x^{2})^{(12 - i)} (\frac{1}{x})^{i}

展开求和

= \frac{12 !}{0 ! ( 12 - 0 ) !} (x^{2})^{12} (\frac{1}{x})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (x^{2})^{11} (\frac{1}{x})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (x^{2})^{10} (\frac{1}{x})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (x^{2})^{9} (\frac{1}{x})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (x^{2})^{8} (\frac{1}{x})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (x^{2})^{7} (\frac{1}{x})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (x^{2})^{6} (\frac{1}{x})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (x^{2})^{5} (\frac{1}{x})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (x^{2})^{4} (\frac{1}{x})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (x^{2})^{3} (\frac{1}{x})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (x^{2})^{2} (\frac{1}{x})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (x^{2})^{1} (\frac{1}{x})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (x^{2})^{0} (\frac{1}{x})^{12}

化简

\frac{12 !}{0 ! ( 12 - 0 ) !} (x^{2})^{12} (\frac{1}{x})^{0} : x^{24}

化简

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{2})^{11} (\frac{1}{x})^{1} : 12 x^{21}

化简

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{2})^{10} (\frac{1}{x})^{2} : 66 x^{18}

化简

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{2})^{9} (\frac{1}{x})^{3} : 220 x^{15}

化简

\frac{12 !}{4 ! ( 12 - 4 ) !} (x^{2})^{8} (\frac{1}{x})^{4} : 495 x^{12}

化简

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{2})^{7} (\frac{1}{x})^{5} : 792 x^{9}

化简

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{2})^{6} (\frac{1}{x})^{6} : 924 x^{6}

化简

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{2})^{5} (\frac{1}{x})^{7} : 792 x^{3}

化简

\frac{12 !}{8 ! ( 12 - 8 ) !} (x^{2})^{4} (\frac{1}{x})^{8} : 495

化简

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{2})^{3} (\frac{1}{x})^{9} : \frac{220}{x ^{3}}

化简

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{2})^{2} (\frac{1}{x})^{10} : \frac{66}{x ^{6}}

化简

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{2})^{1} (\frac{1}{x})^{11} : \frac{12}{x ^{9}}

化简

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

= x^{24} + 12 x^{21} + 66 x^{18} + 220 x^{15} + 495 x^{12} + 792 x^{9} + 924 x^{6} + 792 x^{3} + 495 + \frac{220}{x ^{3}} + \frac{66}{x ^{6}} + \frac{12}{x ^{9}} + \frac{1}{x ^{12}}

流行的例子

展开 (4+x^2)/((4-x^2)^2)

e x p an d \frac{4 + x ^{2}}{( 4 - x ^{2} ) ^{2}}

展开 1/((2n+1)(2n-1))

e x p an d \frac{1}{( 2 n + 1 ) ( 2 n - 1 )}

展开 1/(s(s^2-4))

e x p an d \frac{1}{s ( s ^{2} - 4 )}

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e x p an d (2 - a)^{10}

展开 pi(4y)^2-pi(y^2)^2

e x p an d π (4 y)^{2} - π (y^{2})^{2}