erweitern ((2+x)^{12})/(4096)

Lösung

erweitern

\frac{( 2 + x ) ^{12}}{4096}

\frac{x ^{12}}{4096} + \frac{3 x ^{11}}{512} + \frac{33 x ^{10}}{512} + \frac{55 x ^{9}}{128} + \frac{495 x ^{8}}{256} + \frac{99 x ^{7}}{16} + \frac{231 x ^{6}}{16} + \frac{99 x ^{5}}{4} + \frac{495 x ^{4}}{16} + \frac{55 x ^{3}}{2} + \frac{33 x ^{2}}{2} + 6 x + 1

Schritte zur Lösung

\frac{( 2 + x ) ^{12}}{4096}

(2 + x)^{12} = 4096 + 24576 x + 67584 x^{2} + 112640 x^{3} + 126720 x^{4} + 101376 x^{5} + 59136 x^{6} + 25344 x^{7} + 7920 x^{8} + 1760 x^{9} + 264 x^{10} + 24 x^{11} + x^{12}

= \frac{4096 + 24576 x + 67584 x ^{2} + 112640 x ^{3} + 126720 x ^{4} + 101376 x ^{5} + 59136 x ^{6} + 25344 x ^{7} + 7920 x ^{8} + 1760 x ^{9} + 264 x ^{10} + 24 x ^{11} + x ^{12}}{4096}

Wende Bruchregel an:

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

\frac{4096 + 24576 x + 67584 x ^{2} + 112640 x ^{3} + 126720 x ^{4} + 101376 x ^{5} + 59136 x ^{6} + 25344 x ^{7} + 7920 x ^{8} + 1760 x ^{9} + 264 x ^{10} + 24 x ^{11} + x ^{12}}{4096} = \frac{4096}{4096} + \frac{24576 x}{4096} + \frac{67584 x ^{2}}{4096} + \frac{112640 x ^{3}}{4096} + \frac{126720 x ^{4}}{4096} + \frac{101376 x ^{5}}{4096} + \frac{59136 x ^{6}}{4096} + \frac{25344 x ^{7}}{4096} + \frac{7920 x ^{8}}{4096} + \frac{1760 x ^{9}}{4096} + \frac{264 x ^{10}}{4096} + \frac{24 x ^{11}}{4096} + \frac{x ^{12}}{4096}

= \frac{4096}{4096} + \frac{24576 x}{4096} + \frac{67584 x ^{2}}{4096} + \frac{112640 x ^{3}}{4096} + \frac{126720 x ^{4}}{4096} + \frac{101376 x ^{5}}{4096} + \frac{59136 x ^{6}}{4096} + \frac{25344 x ^{7}}{4096} + \frac{7920 x ^{8}}{4096} + \frac{1760 x ^{9}}{4096} + \frac{264 x ^{10}}{4096} + \frac{24 x ^{11}}{4096} + \frac{x ^{12}}{4096}

Wende Regel an

\frac{a}{a} = 1

\frac{4096}{4096} = 1

= 1 + \frac{24576 x}{4096} + \frac{67584 x ^{2}}{4096} + \frac{112640 x ^{3}}{4096} + \frac{126720 x ^{4}}{4096} + \frac{101376 x ^{5}}{4096} + \frac{59136 x ^{6}}{4096} + \frac{25344 x ^{7}}{4096} + \frac{7920 x ^{8}}{4096} + \frac{1760 x ^{9}}{4096} + \frac{264 x ^{10}}{4096} + \frac{24 x ^{11}}{4096} + \frac{x ^{12}}{4096}

\frac{24576 x}{4096} = 6 x

\frac{67584 x ^{2}}{4096} = \frac{33 x ^{2}}{2}

\frac{112640 x ^{3}}{4096} = \frac{55 x ^{3}}{2}

\frac{126720 x ^{4}}{4096} = \frac{495 x ^{4}}{16}

\frac{101376 x ^{5}}{4096} = \frac{99 x ^{5}}{4}

\frac{59136 x ^{6}}{4096} = \frac{231 x ^{6}}{16}

\frac{25344 x ^{7}}{4096} = \frac{99 x ^{7}}{16}

\frac{7920 x ^{8}}{4096} = \frac{495 x ^{8}}{256}

\frac{1760 x ^{9}}{4096} = \frac{55 x ^{9}}{128}

\frac{264 x ^{10}}{4096} = \frac{33 x ^{10}}{512}

\frac{24 x ^{11}}{4096} = \frac{3 x ^{11}}{512}

= 1 + 6 x + \frac{33 x ^{2}}{2} + \frac{55 x ^{3}}{2} + \frac{495 x ^{4}}{16} + \frac{99 x ^{5}}{4} + \frac{231 x ^{6}}{16} + \frac{99 x ^{7}}{16} + \frac{495 x ^{8}}{256} + \frac{55 x ^{9}}{128} + \frac{33 x ^{10}}{512} + \frac{3 x ^{11}}{512} + \frac{x ^{12}}{4096}

Rewrite in standard form

= \frac{x ^{12}}{4096} + \frac{3 x ^{11}}{512} + \frac{33 x ^{10}}{512} + \frac{55 x ^{9}}{128} + \frac{495 x ^{8}}{256} + \frac{99 x ^{7}}{16} + \frac{231 x ^{6}}{16} + \frac{99 x ^{5}}{4} + \frac{495 x ^{4}}{16} + \frac{55 x ^{3}}{2} + \frac{33 x ^{2}}{2} + 6 x + 1

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