展开 ((x^3-3)^9)/(x^{18)}

解答

展开

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

x^{9} - 27 x^{6} + 324 x^{3} - 2268 + \frac{10206}{x ^{3}} - \frac{30618}{x ^{6}} + \frac{61236}{x ^{9}} - \frac{78732}{x ^{12}} + \frac{59049}{x ^{15}} - \frac{19683}{x ^{18}}

求解步骤

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

(x^{3} - 3)^{9} = x^{27} - 27 x^{24} + 324 x^{21} - 2268 x^{18} + 10206 x^{15} - 30618 x^{12} + 61236 x^{9} - 78732 x^{6} + 59049 x^{3} - 19683

= \frac{x ^{27} - 27 x ^{24} + 324 x ^{21} - 2268 x ^{18} + 10206 x ^{15} - 30618 x ^{12} + 61236 x ^{9} - 78732 x ^{6} + 59049 x ^{3} - 19683}{x ^{18}}

使用分式法则:

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

\frac{x ^{27} - 27 x ^{24} + 324 x ^{21} - 2268 x ^{18} + 10206 x ^{15} - 30618 x ^{12} + 61236 x ^{9} - 78732 x ^{6} + 59049 x ^{3} - 19683}{x ^{18}} = \frac{x ^{27}}{x ^{18}} - \frac{27 x ^{24}}{x ^{18}} + \frac{324 x ^{21}}{x ^{18}} - \frac{2268 x ^{18}}{x ^{18}} + \frac{10206 x ^{15}}{x ^{18}} - \frac{30618 x ^{12}}{x ^{18}} + \frac{61236 x ^{9}}{x ^{18}} - \frac{78732 x ^{6}}{x ^{18}} + \frac{59049 x ^{3}}{x ^{18}} - \frac{19683}{x ^{18}}

= \frac{x ^{27}}{x ^{18}} - \frac{27 x ^{24}}{x ^{18}} + \frac{324 x ^{21}}{x ^{18}} - \frac{2268 x ^{18}}{x ^{18}} + \frac{10206 x ^{15}}{x ^{18}} - \frac{30618 x ^{12}}{x ^{18}} + \frac{61236 x ^{9}}{x ^{18}} - \frac{78732 x ^{6}}{x ^{18}} + \frac{59049 x ^{3}}{x ^{18}} - \frac{19683}{x ^{18}}

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

\frac{27 x ^{24}}{x ^{18}} = 27 x^{6}

\frac{324 x ^{21}}{x ^{18}} = 324 x^{3}

\frac{2268 x ^{18}}{x ^{18}} = 2268

\frac{10206 x ^{15}}{x ^{18}} = \frac{10206}{x ^{3}}

\frac{30618 x ^{12}}{x ^{18}} = \frac{30618}{x ^{6}}

\frac{61236 x ^{9}}{x ^{18}} = \frac{61236}{x ^{9}}

\frac{78732 x ^{6}}{x ^{18}} = \frac{78732}{x ^{12}}

\frac{59049 x ^{3}}{x ^{18}} = \frac{59049}{x ^{15}}

= x^{9} - 27 x^{6} + 324 x^{3} - 2268 + \frac{10206}{x ^{3}} - \frac{30618}{x ^{6}} + \frac{61236}{x ^{9}} - \frac{78732}{x ^{12}} + \frac{59049}{x ^{15}} - \frac{19683}{x ^{18}}

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