展開する ((x^3-2)^9)/(x^{18)}

解

展開する

\frac{( x ^{3} - 2 ) ^{9}}{x ^{18}}

x^{9} - 18 x^{6} + 144 x^{3} - 672 + \frac{2016}{x ^{3}} - \frac{4032}{x ^{6}} + \frac{5376}{x ^{9}} - \frac{4608}{x ^{12}} + \frac{2304}{x ^{15}} - \frac{512}{x ^{18}}

解答ステップ

\frac{( x ^{3} - 2 ) ^{9}}{x ^{18}}

(x^{3} - 2)^{9} = x^{27} - 18 x^{24} + 144 x^{21} - 672 x^{18} + 2016 x^{15} - 4032 x^{12} + 5376 x^{9} - 4608 x^{6} + 2304 x^{3} - 512

= \frac{x ^{27} - 18 x ^{24} + 144 x ^{21} - 672 x ^{18} + 2016 x ^{15} - 4032 x ^{12} + 5376 x ^{9} - 4608 x ^{6} + 2304 x ^{3} - 512}{x ^{18}}

分数の規則を適用する:

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

\frac{x ^{27} - 18 x ^{24} + 144 x ^{21} - 672 x ^{18} + 2016 x ^{15} - 4032 x ^{12} + 5376 x ^{9} - 4608 x ^{6} + 2304 x ^{3} - 512}{x ^{18}} = \frac{x ^{27}}{x ^{18}} - \frac{18 x ^{24}}{x ^{18}} + \frac{144 x ^{21}}{x ^{18}} - \frac{672 x ^{18}}{x ^{18}} + \frac{2016 x ^{15}}{x ^{18}} - \frac{4032 x ^{12}}{x ^{18}} + \frac{5376 x ^{9}}{x ^{18}} - \frac{4608 x ^{6}}{x ^{18}} + \frac{2304 x ^{3}}{x ^{18}} - \frac{512}{x ^{18}}

= \frac{x ^{27}}{x ^{18}} - \frac{18 x ^{24}}{x ^{18}} + \frac{144 x ^{21}}{x ^{18}} - \frac{672 x ^{18}}{x ^{18}} + \frac{2016 x ^{15}}{x ^{18}} - \frac{4032 x ^{12}}{x ^{18}} + \frac{5376 x ^{9}}{x ^{18}} - \frac{4608 x ^{6}}{x ^{18}} + \frac{2304 x ^{3}}{x ^{18}} - \frac{512}{x ^{18}}

\frac{x ^{27}}{x ^{18}} = x^{9}

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

\frac{144 x ^{21}}{x ^{18}} = 144 x^{3}

\frac{672 x ^{18}}{x ^{18}} = 672

\frac{2016 x ^{15}}{x ^{18}} = \frac{2016}{x ^{3}}

\frac{4032 x ^{12}}{x ^{18}} = \frac{4032}{x ^{6}}

\frac{5376 x ^{9}}{x ^{18}} = \frac{5376}{x ^{9}}

\frac{4608 x ^{6}}{x ^{18}} = \frac{4608}{x ^{12}}

\frac{2304 x ^{3}}{x ^{18}} = \frac{2304}{x ^{15}}

= x^{9} - 18 x^{6} + 144 x^{3} - 672 + \frac{2016}{x ^{3}} - \frac{4032}{x ^{6}} + \frac{5376}{x ^{9}} - \frac{4608}{x ^{12}} + \frac{2304}{x ^{15}} - \frac{512}{x ^{18}}