展開する (1+3/x)^2(1+3/4 x)^6

解

展開する

(1 + \frac{3}{x})^{2} (1 + \frac{3}{4} x)^{6}

\frac{x ^{2} + 6 x + 9}{x ^{2}} + \frac{9 ( x ^{2} + 6 x + 9 )}{2 x} + \frac{135 x ^{3} + 945 x ^{2} + 2025 x + 1215}{16} + \frac{1215 ( x ^{2} + 6 x + 9 ) x ^{2}}{256} + \frac{729 ( x ^{2} + 6 x + 9 ) x ^{3}}{512} + \frac{729 ( x ^{2} + 6 x + 9 ) x ^{4}}{4096}

解答ステップ

(1 + \frac{3}{x})^{2} (1 + \frac{3}{4} x)^{6}

拡張

(1 + \frac{3}{x})^{2} : \frac{x ^{2} + 6 x + 9}{x ^{2}}

= \frac{x ^{2} + 6 x + 9}{x ^{2}} (1 + \frac{3}{4} x)^{6}

拡張

(1 + \frac{3}{4} x)^{6} : 1 + \frac{9 x}{2} + \frac{135 x ^{2}}{16} + \frac{135 x ^{3}}{16} + \frac{1215 x ^{4}}{256} + \frac{729 x ^{5}}{512} + \frac{729 x ^{6}}{4096}

= \frac{x ^{2} + 6 x + 9}{x ^{2}} (1 + \frac{9 x}{2} + \frac{135 x ^{2}}{16} + \frac{135 x ^{3}}{16} + \frac{1215 x ^{4}}{256} + \frac{729 x ^{5}}{512} + \frac{729 x ^{6}}{4096})

括弧を分配する

= \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot 1 + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{9 x}{2} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{135 x ^{2}}{16} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{135 x ^{3}}{16} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{1215 x ^{4}}{256} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{729 x ^{5}}{512} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{729 x ^{6}}{4096}

簡素化

\frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot 1 + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{9 x}{2} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{135 x ^{2}}{16} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{135 x ^{3}}{16} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{1215 x ^{4}}{256} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{729 x ^{5}}{512} + \frac{x ^{2} + 6 x + 9}{x ^{2}} \cdot \frac{729 x ^{6}}{4096} : \frac{x ^{2} + 6 x + 9}{x ^{2}} + \frac{9 ( x ^{2} + 6 x + 9 )}{2 x} + \frac{135 x ^{3} + 945 x ^{2} + 2025 x + 1215}{16} + \frac{1215 ( x ^{2} + 6 x + 9 ) x ^{2}}{256} + \frac{729 ( x ^{2} + 6 x + 9 ) x ^{3}}{512} + \frac{729 ( x ^{2} + 6 x + 9 ) x ^{4}}{4096}

= \frac{x ^{2} + 6 x + 9}{x ^{2}} + \frac{9 ( x ^{2} + 6 x + 9 )}{2 x} + \frac{135 x ^{3} + 945 x ^{2} + 2025 x + 1215}{16} + \frac{1215 ( x ^{2} + 6 x + 9 ) x ^{2}}{256} + \frac{729 ( x ^{2} + 6 x + 9 ) x ^{3}}{512} + \frac{729 ( x ^{2} + 6 x + 9 ) x ^{4}}{4096}