展開する ((x^3-a)^9)/(x^{18)}

解

展開する

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

x^{9} - 9 a x^{6} + 36 a^{2} x^{3} - 84 a^{3} + \frac{126 a ^{4}}{x ^{3}} - \frac{126 a ^{5}}{x ^{6}} + \frac{84 a ^{6}}{x ^{9}} - \frac{36 a ^{7}}{x ^{12}} + \frac{9 a ^{8}}{x ^{15}} - \frac{a ^{9}}{x ^{18}}

解答ステップ

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

(x^{3} - a)^{9} = x^{27} - 9 a x^{24} + 36 a^{2} x^{21} - 84 a^{3} x^{18} + 126 a^{4} x^{15} - 126 a^{5} x^{12} + 84 a^{6} x^{9} - 36 a^{7} x^{6} + 9 a^{8} x^{3} - a^{9}

= \frac{x ^{27} - 9 a x ^{24} + 36 a ^{2} x ^{21} - 84 a ^{3} x ^{18} + 126 a ^{4} x ^{15} - 126 a ^{5} x ^{12} + 84 a ^{6} x ^{9} - 36 a ^{7} x ^{6} + 9 a ^{8} x ^{3} - a ^{9}}{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} - 9 a x ^{24} + 36 a ^{2} x ^{21} - 84 a ^{3} x ^{18} + 126 a ^{4} x ^{15} - 126 a ^{5} x ^{12} + 84 a ^{6} x ^{9} - 36 a ^{7} x ^{6} + 9 a ^{8} x ^{3} - a ^{9}}{x ^{18}} = \frac{x ^{27}}{x ^{18}} - \frac{9 a x ^{24}}{x ^{18}} + \frac{36 a ^{2} x ^{21}}{x ^{18}} - \frac{84 a ^{3} x ^{18}}{x ^{18}} + \frac{126 a ^{4} x ^{15}}{x ^{18}} - \frac{126 a ^{5} x ^{12}}{x ^{18}} + \frac{84 a ^{6} x ^{9}}{x ^{18}} - \frac{36 a ^{7} x ^{6}}{x ^{18}} + \frac{9 a ^{8} x ^{3}}{x ^{18}} - \frac{a ^{9}}{x ^{18}}

= \frac{x ^{27}}{x ^{18}} - \frac{9 a x ^{24}}{x ^{18}} + \frac{36 a ^{2} x ^{21}}{x ^{18}} - \frac{84 a ^{3} x ^{18}}{x ^{18}} + \frac{126 a ^{4} x ^{15}}{x ^{18}} - \frac{126 a ^{5} x ^{12}}{x ^{18}} + \frac{84 a ^{6} x ^{9}}{x ^{18}} - \frac{36 a ^{7} x ^{6}}{x ^{18}} + \frac{9 a ^{8} x ^{3}}{x ^{18}} - \frac{a ^{9}}{x ^{18}}

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

\frac{9 a x ^{24}}{x ^{18}} = 9 a x^{6}

\frac{36 a ^{2} x ^{21}}{x ^{18}} = 36 a^{2} x^{3}

\frac{84 a ^{3} x ^{18}}{x ^{18}} = 84 a^{3}

\frac{126 a ^{4} x ^{15}}{x ^{18}} = \frac{126 a ^{4}}{x ^{3}}

\frac{126 a ^{5} x ^{12}}{x ^{18}} = \frac{126 a ^{5}}{x ^{6}}

\frac{84 a ^{6} x ^{9}}{x ^{18}} = \frac{84 a ^{6}}{x ^{9}}

\frac{36 a ^{7} x ^{6}}{x ^{18}} = \frac{36 a ^{7}}{x ^{12}}

\frac{9 a ^{8} x ^{3}}{x ^{18}} = \frac{9 a ^{8}}{x ^{15}}

= x^{9} - 9 a x^{6} + 36 a^{2} x^{3} - 84 a^{3} + \frac{126 a ^{4}}{x ^{3}} - \frac{126 a ^{5}}{x ^{6}} + \frac{84 a ^{6}}{x ^{9}} - \frac{36 a ^{7}}{x ^{12}} + \frac{9 a ^{8}}{x ^{15}} - \frac{a ^{9}}{x ^{18}}