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受欢迎的代数 >

展开 (x+2/x)^8

解答

展开

(x + \frac{2}{x})^{8}

解答

x^{8} + 16 x^{6} + 112 x^{4} + 448 x^{2} + 1120 + \frac{1792}{x ^{2}} + \frac{1792}{x ^{4}} + \frac{1024}{x ^{6}} + \frac{256}{x ^{8}}

求解步骤

(x + \frac{2}{x})^{8}

使用二项式定理:

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

a = x, b = \frac{2}{x}

= i = 0 \sum 8 (i 8) x^{(8 - i)} (\frac{2}{x})^{i}

展开求和

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

化简

\frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} (\frac{2}{x})^{0} : x^{8}

化简

\frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} (\frac{2}{x})^{1} : 16 x^{6}

化简

\frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} (\frac{2}{x})^{2} : 112 x^{4}

化简

\frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} (\frac{2}{x})^{3} : 448 x^{2}

化简

\frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} (\frac{2}{x})^{4} : 1120

化简

\frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} (\frac{2}{x})^{5} : \frac{1792}{x ^{2}}

化简

\frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} (\frac{2}{x})^{6} : \frac{1792}{x ^{4}}

化简

\frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} (\frac{2}{x})^{7} : \frac{1024}{x ^{6}}

化简

\frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} (\frac{2}{x})^{8} : \frac{256}{x ^{8}}

= x^{8} + 16 x^{6} + 112 x^{4} + 448 x^{2} + 1120 + \frac{1792}{x ^{2}} + \frac{1792}{x ^{4}} + \frac{1024}{x ^{6}} + \frac{256}{x ^{8}}

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