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Beliebt Algebra >

erweitern (x/2+1/x)^{12}

Lösung

erweitern

(\frac{x}{2} + \frac{1}{x})^{12}

Lösung

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

Schritte zur Lösung

(\frac{x}{2} + \frac{1}{x})^{12}

Wende Binomialsetz an:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = \frac{x}{2}, b = \frac{1}{x}

= i = 0 \sum 12 (i 12) (\frac{x}{2})^{(12 - i)} (\frac{1}{x})^{i}

Erweitere Summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{x}{2})^{12} (\frac{1}{x})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{x}{2})^{11} (\frac{1}{x})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{x}{2})^{10} (\frac{1}{x})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{x}{2})^{9} (\frac{1}{x})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{x}{2})^{8} (\frac{1}{x})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{x}{2})^{7} (\frac{1}{x})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{x}{2})^{6} (\frac{1}{x})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{x}{2})^{5} (\frac{1}{x})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{x}{2})^{4} (\frac{1}{x})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{x}{2})^{3} (\frac{1}{x})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{x}{2})^{2} (\frac{1}{x})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{x}{2})^{1} (\frac{1}{x})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{x}{2})^{0} (\frac{1}{x})^{12}

Vereinfache

\frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{x}{2})^{12} (\frac{1}{x})^{0} : \frac{x ^{12}}{4096}

Vereinfache

\frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{x}{2})^{11} (\frac{1}{x})^{1} : \frac{3 x ^{10}}{512}

Vereinfache

\frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{x}{2})^{10} (\frac{1}{x})^{2} : \frac{33 x ^{8}}{512}

Vereinfache

\frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{x}{2})^{9} (\frac{1}{x})^{3} : \frac{55 x ^{6}}{128}

Vereinfache

\frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{x}{2})^{8} (\frac{1}{x})^{4} : \frac{495 x ^{4}}{256}

Vereinfache

\frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{x}{2})^{7} (\frac{1}{x})^{5} : \frac{99 x ^{2}}{16}

Vereinfache

\frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{x}{2})^{6} (\frac{1}{x})^{6} : \frac{231}{16}

Vereinfache

\frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{x}{2})^{5} (\frac{1}{x})^{7} : \frac{99}{4 x ^{2}}

Vereinfache

\frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{x}{2})^{4} (\frac{1}{x})^{8} : \frac{495}{16 x ^{4}}

Vereinfache

\frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{x}{2})^{3} (\frac{1}{x})^{9} : \frac{55}{2 x ^{6}}

Vereinfache

\frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{x}{2})^{2} (\frac{1}{x})^{10} : \frac{33}{2 x ^{8}}

Vereinfache

\frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{x}{2})^{1} (\frac{1}{x})^{11} : \frac{6}{x ^{10}}

Vereinfache

\frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{x}{2})^{0} (\frac{1}{x})^{12} : \frac{1}{x ^{12}}

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

Beliebte Beispiele

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