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受欢迎的代数 >

展开 (2x+1/(x^2))^6

解答

展开

(2 x + \frac{1}{x ^{2}})^{6}

解答

64 x^{6} + 192 x^{3} + 240 + \frac{160}{x ^{3}} + \frac{60}{x ^{6}} + \frac{12}{x ^{9}} + \frac{1}{x ^{12}}

求解步骤

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

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

展开求和

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

化简

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

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} (2 x)^{5} (\frac{1}{x ^{2}})^{1} : 192 x^{3}

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} (2 x)^{4} (\frac{1}{x ^{2}})^{2} : 240

化简

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

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} (2 x)^{2} (\frac{1}{x ^{2}})^{4} : \frac{60}{x ^{6}}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} (2 x)^{1} (\frac{1}{x ^{2}})^{5} : \frac{12}{x ^{9}}

化简

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

= 64 x^{6} + 192 x^{3} + 240 + \frac{160}{x ^{3}} + \frac{60}{x ^{6}} + \frac{12}{x ^{9}} + \frac{1}{x ^{12}}

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