展开 ((2x+1)/x)^7

解答

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(\frac{2 x + 1}{x})^{7}

128 + \frac{448}{x} + \frac{672}{x ^{2}} + \frac{560}{x ^{3}} + \frac{280}{x ^{4}} + \frac{84}{x ^{5}} + \frac{14}{x ^{6}} + \frac{1}{x ^{7}}

求解步骤

(\frac{2 x + 1}{x})^{7}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 x + 1 ) ^{7}}{x ^{7}}

(2 x + 1)^{7} = 128 x^{7} + 448 x^{6} + 672 x^{5} + 560 x^{4} + 280 x^{3} + 84 x^{2} + 14 x + 1

= \frac{128 x ^{7} + 448 x ^{6} + 672 x ^{5} + 560 x ^{4} + 280 x ^{3} + 84 x ^{2} + 14 x + 1}{x ^{7}}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{128 x ^{7} + 448 x ^{6} + 672 x ^{5} + 560 x ^{4} + 280 x ^{3} + 84 x ^{2} + 14 x + 1}{x ^{7}} = \frac{128 x ^{7}}{x ^{7}} + \frac{448 x ^{6}}{x ^{7}} + \frac{672 x ^{5}}{x ^{7}} + \frac{560 x ^{4}}{x ^{7}} + \frac{280 x ^{3}}{x ^{7}} + \frac{84 x ^{2}}{x ^{7}} + \frac{14 x}{x ^{7}} + \frac{1}{x ^{7}}

= \frac{128 x ^{7}}{x ^{7}} + \frac{448 x ^{6}}{x ^{7}} + \frac{672 x ^{5}}{x ^{7}} + \frac{560 x ^{4}}{x ^{7}} + \frac{280 x ^{3}}{x ^{7}} + \frac{84 x ^{2}}{x ^{7}} + \frac{14 x}{x ^{7}} + \frac{1}{x ^{7}}

\frac{128 x ^{7}}{x ^{7}} = 128

\frac{448 x ^{6}}{x ^{7}} = \frac{448}{x}

\frac{672 x ^{5}}{x ^{7}} = \frac{672}{x ^{2}}

\frac{560 x ^{4}}{x ^{7}} = \frac{560}{x ^{3}}

\frac{280 x ^{3}}{x ^{7}} = \frac{280}{x ^{4}}

\frac{84 x ^{2}}{x ^{7}} = \frac{84}{x ^{5}}

\frac{14 x}{x ^{7}} = \frac{14}{x ^{6}}

= 128 + \frac{448}{x} + \frac{672}{x ^{2}} + \frac{560}{x ^{3}} + \frac{280}{x ^{4}} + \frac{84}{x ^{5}} + \frac{14}{x ^{6}} + \frac{1}{x ^{7}}

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