展開する 1/(x^3)(1/(3x^2)-x/2)^9

解

展開する

\frac{1}{x ^{3}} (\frac{1}{3 x ^{2}} - \frac{x}{2})^{9}

\frac{1}{19683 x ^{21}} - \frac{1}{1458 x ^{18}} + \frac{1}{243 x ^{15}} - \frac{7}{486 x ^{12}} + \frac{7}{216 x ^{9}} - \frac{7}{144 x ^{6}} + \frac{7}{144 x ^{3}} - \frac{1}{32} + \frac{3 x ^{3}}{256} - \frac{x ^{6}}{512}

解答ステップ

\frac{1}{x ^{3}} (\frac{1}{3 x ^{2}} - \frac{x}{2})^{9}

拡張

(\frac{1}{3 x ^{2}} - \frac{x}{2})^{9} : \frac{1}{19683 x ^{18}} - \frac{1}{1458 x ^{15}} + \frac{1}{243 x ^{12}} - \frac{7}{486 x ^{9}} + \frac{7}{216 x ^{6}} - \frac{7}{144 x ^{3}} + \frac{7}{144} - \frac{x ^{3}}{32} + \frac{3 x ^{6}}{256} - \frac{x ^{9}}{512}

= \frac{1}{x ^{3}} (\frac{1}{19683 x ^{18}} - \frac{1}{1458 x ^{15}} + \frac{1}{243 x ^{12}} - \frac{7}{486 x ^{9}} + \frac{7}{216 x ^{6}} - \frac{7}{144 x ^{3}} + \frac{7}{144} - \frac{x ^{3}}{32} + \frac{3 x ^{6}}{256} - \frac{x ^{9}}{512})

括弧を分配する

= \frac{1}{x ^{3}} \cdot \frac{1}{19683 x ^{18}} + \frac{1}{x ^{3}} (- \frac{1}{1458 x ^{15}}) + \frac{1}{x ^{3}} \cdot \frac{1}{243 x ^{12}} + \frac{1}{x ^{3}} (- \frac{7}{486 x ^{9}}) + \frac{1}{x ^{3}} \cdot \frac{7}{216 x ^{6}} + \frac{1}{x ^{3}} (- \frac{7}{144 x ^{3}}) + \frac{1}{x ^{3}} \cdot \frac{7}{144} + \frac{1}{x ^{3}} (- \frac{x ^{3}}{32}) + \frac{1}{x ^{3}} \cdot \frac{3 x ^{6}}{256} + \frac{1}{x ^{3}} (- \frac{x ^{9}}{512})

簡素化

\frac{1}{x ^{3}} \cdot \frac{1}{19683 x ^{18}} + \frac{1}{x ^{3}} (- \frac{1}{1458 x ^{15}}) + \frac{1}{x ^{3}} \cdot \frac{1}{243 x ^{12}} + \frac{1}{x ^{3}} (- \frac{7}{486 x ^{9}}) + \frac{1}{x ^{3}} \cdot \frac{7}{216 x ^{6}} + \frac{1}{x ^{3}} (- \frac{7}{144 x ^{3}}) + \frac{1}{x ^{3}} \cdot \frac{7}{144} + \frac{1}{x ^{3}} (- \frac{x ^{3}}{32}) + \frac{1}{x ^{3}} \cdot \frac{3 x ^{6}}{256} + \frac{1}{x ^{3}} (- \frac{x ^{9}}{512}) : \frac{1}{19683 x ^{21}} - \frac{1}{1458 x ^{18}} + \frac{1}{243 x ^{15}} - \frac{7}{486 x ^{12}} + \frac{7}{216 x ^{9}} - \frac{7}{144 x ^{6}} + \frac{7}{144 x ^{3}} - \frac{1}{32} + \frac{3 x ^{3}}{256} - \frac{x ^{6}}{512}

= \frac{1}{19683 x ^{21}} - \frac{1}{1458 x ^{18}} + \frac{1}{243 x ^{15}} - \frac{7}{486 x ^{12}} + \frac{7}{216 x ^{9}} - \frac{7}{144 x ^{6}} + \frac{7}{144 x ^{3}} - \frac{1}{32} + \frac{3 x ^{3}}{256} - \frac{x ^{6}}{512}