展開する ((x^2-1)/(x^2))^{12}

解

展開する

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

1 - \frac{12}{x ^{2}} + \frac{66}{x ^{4}} - \frac{220}{x ^{6}} + \frac{495}{x ^{8}} - \frac{792}{x ^{10}} + \frac{924}{x ^{12}} - \frac{792}{x ^{14}} + \frac{495}{x ^{16}} - \frac{220}{x ^{18}} + \frac{66}{x ^{20}} - \frac{12}{x ^{22}} + \frac{1}{x ^{24}}

解答ステップ

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

指数の規則を適用する:

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

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

(x^{2})^{12} : x^{24}

= \frac{( x ^{2} - 1 ) ^{12}}{x ^{24}}

(x^{2} - 1)^{12} = x^{24} - 12 x^{22} + 66 x^{20} - 220 x^{18} + 495 x^{16} - 792 x^{14} + 924 x^{12} - 792 x^{10} + 495 x^{8} - 220 x^{6} + 66 x^{4} - 12 x^{2} + 1

= \frac{x ^{24} - 12 x ^{22} + 66 x ^{20} - 220 x ^{18} + 495 x ^{16} - 792 x ^{14} + 924 x ^{12} - 792 x ^{10} + 495 x ^{8} - 220 x ^{6} + 66 x ^{4} - 12 x ^{2} + 1}{x ^{24}}

分数の規則を適用する:

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

\frac{x ^{24} - 12 x ^{22} + 66 x ^{20} - 220 x ^{18} + 495 x ^{16} - 792 x ^{14} + 924 x ^{12} - 792 x ^{10} + 495 x ^{8} - 220 x ^{6} + 66 x ^{4} - 12 x ^{2} + 1}{x ^{24}} = \frac{x ^{24}}{x ^{24}} - \frac{12 x ^{22}}{x ^{24}} + \frac{66 x ^{20}}{x ^{24}} - \frac{220 x ^{18}}{x ^{24}} + \frac{495 x ^{16}}{x ^{24}} - \frac{792 x ^{14}}{x ^{24}} + \frac{924 x ^{12}}{x ^{24}} - \frac{792 x ^{10}}{x ^{24}} + \frac{495 x ^{8}}{x ^{24}} - \frac{220 x ^{6}}{x ^{24}} + \frac{66 x ^{4}}{x ^{24}} - \frac{12 x ^{2}}{x ^{24}} + \frac{1}{x ^{24}}

= \frac{x ^{24}}{x ^{24}} - \frac{12 x ^{22}}{x ^{24}} + \frac{66 x ^{20}}{x ^{24}} - \frac{220 x ^{18}}{x ^{24}} + \frac{495 x ^{16}}{x ^{24}} - \frac{792 x ^{14}}{x ^{24}} + \frac{924 x ^{12}}{x ^{24}} - \frac{792 x ^{10}}{x ^{24}} + \frac{495 x ^{8}}{x ^{24}} - \frac{220 x ^{6}}{x ^{24}} + \frac{66 x ^{4}}{x ^{24}} - \frac{12 x ^{2}}{x ^{24}} + \frac{1}{x ^{24}}

規則を適用

\frac{a}{a} = 1

\frac{x ^{24}}{x ^{24}} = 1

= 1 - \frac{12 x ^{22}}{x ^{24}} + \frac{66 x ^{20}}{x ^{24}} - \frac{220 x ^{18}}{x ^{24}} + \frac{495 x ^{16}}{x ^{24}} - \frac{792 x ^{14}}{x ^{24}} + \frac{924 x ^{12}}{x ^{24}} - \frac{792 x ^{10}}{x ^{24}} + \frac{495 x ^{8}}{x ^{24}} - \frac{220 x ^{6}}{x ^{24}} + \frac{66 x ^{4}}{x ^{24}} - \frac{12 x ^{2}}{x ^{24}} + \frac{1}{x ^{24}}

\frac{12 x ^{22}}{x ^{24}} = \frac{12}{x ^{2}}

\frac{66 x ^{20}}{x ^{24}} = \frac{66}{x ^{4}}

\frac{220 x ^{18}}{x ^{24}} = \frac{220}{x ^{6}}

\frac{495 x ^{16}}{x ^{24}} = \frac{495}{x ^{8}}

\frac{792 x ^{14}}{x ^{24}} = \frac{792}{x ^{10}}

\frac{924 x ^{12}}{x ^{24}} = \frac{924}{x ^{12}}

\frac{792 x ^{10}}{x ^{24}} = \frac{792}{x ^{14}}

\frac{495 x ^{8}}{x ^{24}} = \frac{495}{x ^{16}}

\frac{220 x ^{6}}{x ^{24}} = \frac{220}{x ^{18}}

\frac{66 x ^{4}}{x ^{24}} = \frac{66}{x ^{20}}

\frac{12 x ^{2}}{x ^{24}} = \frac{12}{x ^{22}}

= 1 - \frac{12}{x ^{2}} + \frac{66}{x ^{4}} - \frac{220}{x ^{6}} + \frac{495}{x ^{8}} - \frac{792}{x ^{10}} + \frac{924}{x ^{12}} - \frac{792}{x ^{14}} + \frac{495}{x ^{16}} - \frac{220}{x ^{18}} + \frac{66}{x ^{20}} - \frac{12}{x ^{22}} + \frac{1}{x ^{24}}