展開する ((x^3-4)^6)/(64x^6)

解

展開する

\frac{( x ^{3} - 4 ) ^{6}}{64 x ^{6}}

\frac{x ^{12}}{64} - \frac{3 x ^{9}}{8} + \frac{15 x ^{6}}{4} - 20 x^{3} + 60 - \frac{96}{x ^{3}} + \frac{64}{x ^{6}}

解答ステップ

\frac{( x ^{3} - 4 ) ^{6}}{64 x ^{6}}

(x^{3} - 4)^{6} = x^{18} - 24 x^{15} + 240 x^{12} - 1280 x^{9} + 3840 x^{6} - 6144 x^{3} + 4096

= \frac{x ^{18} - 24 x ^{15} + 240 x ^{12} - 1280 x ^{9} + 3840 x ^{6} - 6144 x ^{3} + 4096}{64 x ^{6}}

分数の規則を適用する:

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

\frac{x ^{18} - 24 x ^{15} + 240 x ^{12} - 1280 x ^{9} + 3840 x ^{6} - 6144 x ^{3} + 4096}{64 x ^{6}} = \frac{x ^{18}}{64 x ^{6}} - \frac{24 x ^{15}}{64 x ^{6}} + \frac{240 x ^{12}}{64 x ^{6}} - \frac{1280 x ^{9}}{64 x ^{6}} + \frac{3840 x ^{6}}{64 x ^{6}} - \frac{6144 x ^{3}}{64 x ^{6}} + \frac{4096}{64 x ^{6}}

= \frac{x ^{18}}{64 x ^{6}} - \frac{24 x ^{15}}{64 x ^{6}} + \frac{240 x ^{12}}{64 x ^{6}} - \frac{1280 x ^{9}}{64 x ^{6}} + \frac{3840 x ^{6}}{64 x ^{6}} - \frac{6144 x ^{3}}{64 x ^{6}} + \frac{4096}{64 x ^{6}}

\frac{x ^{18}}{64 x ^{6}} = \frac{x ^{12}}{64}

\frac{24 x ^{15}}{64 x ^{6}} = \frac{3 x ^{9}}{8}

\frac{240 x ^{12}}{64 x ^{6}} = \frac{15 x ^{6}}{4}

\frac{1280 x ^{9}}{64 x ^{6}} = 20 x^{3}

\frac{3840 x ^{6}}{64 x ^{6}} = 60

\frac{6144 x ^{3}}{64 x ^{6}} = \frac{96}{x ^{3}}

\frac{4096}{64 x ^{6}} = \frac{64}{x ^{6}}

= \frac{x ^{12}}{64} - \frac{3 x ^{9}}{8} + \frac{15 x ^{6}}{4} - 20 x^{3} + 60 - \frac{96}{x ^{3}} + \frac{64}{x ^{6}}