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Beliebt Algebra >

erweitern ((x^2)/2+4/x)^6

Lösung

erweitern

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

Lösung

\frac{x ^{12}}{64} + \frac{3 x ^{9}}{4} + 15 x^{6} + 160 x^{3} + 960 + \frac{3072}{x ^{3}} + \frac{4096}{x ^{6}}

Schritte zur Lösung

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

Wende Binomialsetz an:

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

a = \frac{x ^{2}}{2}, b = \frac{4}{x}

= i = 0 \sum 6 (i 6) (\frac{x ^{2}}{2})^{(6 - i)} (\frac{4}{x})^{i}

Erweitere Summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{x ^{2}}{2})^{6} (\frac{4}{x})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{x ^{2}}{2})^{5} (\frac{4}{x})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{x ^{2}}{2})^{4} (\frac{4}{x})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (\frac{x ^{2}}{2})^{3} (\frac{4}{x})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{x ^{2}}{2})^{2} (\frac{4}{x})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (\frac{x ^{2}}{2})^{1} (\frac{4}{x})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{x ^{2}}{2})^{0} (\frac{4}{x})^{6}

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{x ^{2}}{2})^{6} (\frac{4}{x})^{0} : \frac{x ^{12}}{64}

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{x ^{2}}{2})^{5} (\frac{4}{x})^{1} : \frac{3 x ^{9}}{4}

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{x ^{2}}{2})^{4} (\frac{4}{x})^{2} : 15 x^{6}

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} (\frac{x ^{2}}{2})^{3} (\frac{4}{x})^{3} : 160 x^{3}

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{x ^{2}}{2})^{2} (\frac{4}{x})^{4} : 960

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} (\frac{x ^{2}}{2})^{1} (\frac{4}{x})^{5} : \frac{3072}{x ^{3}}

Vereinfache

\frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{x ^{2}}{2})^{0} (\frac{4}{x})^{6} : \frac{4096}{x ^{6}}

= \frac{x ^{12}}{64} + \frac{3 x ^{9}}{4} + 15 x^{6} + 160 x^{3} + 960 + \frac{3072}{x ^{3}} + \frac{4096}{x ^{6}}

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