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受欢迎的代数 >

展开 (2x+1/x)^{12}

解答

展开

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

解答

4096 x^{12} + 24576 x^{10} + 67584 x^{8} + 112640 x^{6} + 126720 x^{4} + 101376 x^{2} + 59136 + \frac{25344}{x ^{2}} + \frac{7920}{x ^{4}} + \frac{1760}{x ^{6}} + \frac{264}{x ^{8}} + \frac{24}{x ^{10}} + \frac{1}{x ^{12}}

求解步骤

(2 x + \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 = 2 x, b = \frac{1}{x}

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

展开求和

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

化简

\frac{12 !}{0 ! ( 12 - 0 ) !} (2 x)^{12} (\frac{1}{x})^{0} : 4096 x^{12}

化简

\frac{12 !}{1 ! ( 12 - 1 ) !} (2 x)^{11} (\frac{1}{x})^{1} : 24576 x^{10}

化简

\frac{12 !}{2 ! ( 12 - 2 ) !} (2 x)^{10} (\frac{1}{x})^{2} : 67584 x^{8}

化简

\frac{12 !}{3 ! ( 12 - 3 ) !} (2 x)^{9} (\frac{1}{x})^{3} : 112640 x^{6}

化简

\frac{12 !}{4 ! ( 12 - 4 ) !} (2 x)^{8} (\frac{1}{x})^{4} : 126720 x^{4}

化简

\frac{12 !}{5 ! ( 12 - 5 ) !} (2 x)^{7} (\frac{1}{x})^{5} : 101376 x^{2}

化简

\frac{12 !}{6 ! ( 12 - 6 ) !} (2 x)^{6} (\frac{1}{x})^{6} : 59136

化简

\frac{12 !}{7 ! ( 12 - 7 ) !} (2 x)^{5} (\frac{1}{x})^{7} : \frac{25344}{x ^{2}}

化简

\frac{12 !}{8 ! ( 12 - 8 ) !} (2 x)^{4} (\frac{1}{x})^{8} : \frac{7920}{x ^{4}}

化简

\frac{12 !}{9 ! ( 12 - 9 ) !} (2 x)^{3} (\frac{1}{x})^{9} : \frac{1760}{x ^{6}}

化简

\frac{12 !}{10 ! ( 12 - 10 ) !} (2 x)^{2} (\frac{1}{x})^{10} : \frac{264}{x ^{8}}

化简

\frac{12 !}{11 ! ( 12 - 11 ) !} (2 x)^{1} (\frac{1}{x})^{11} : \frac{24}{x ^{10}}

化简

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

= 4096 x^{12} + 24576 x^{10} + 67584 x^{8} + 112640 x^{6} + 126720 x^{4} + 101376 x^{2} + 59136 + \frac{25344}{x ^{2}} + \frac{7920}{x ^{4}} + \frac{1760}{x ^{6}} + \frac{264}{x ^{8}} + \frac{24}{x ^{10}} + \frac{1}{x ^{12}}

流行的例子

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展开 2(x+Δx)^3

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