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Beliebt Algebra >

erweitern ((x^2)/2+1/x)^6

Lösung

erweitern

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

Lösung

\frac{x ^{12}}{64} + \frac{3 x ^{9}}{16} + \frac{15 x ^{6}}{16} + \frac{5 x ^{3}}{2} + \frac{15}{4} + \frac{3}{x ^{3}} + \frac{1}{x ^{6}}

Schritte zur Lösung

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

= i = 0 \sum 6 (i 6) (\frac{x ^{2}}{2})^{(6 - i)} (\frac{1}{x})^{i}

Erweitere Summation

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

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{x ^{2}}{2})^{6} (\frac{1}{x})^{0} : \frac{x ^{12}}{64}

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{x ^{2}}{2})^{5} (\frac{1}{x})^{1} : \frac{3 x ^{9}}{16}

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{x ^{2}}{2})^{4} (\frac{1}{x})^{2} : \frac{15 x ^{6}}{16}

Vereinfache

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

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{x ^{2}}{2})^{2} (\frac{1}{x})^{4} : \frac{15}{4}

Vereinfache

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

Vereinfache

\frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{x ^{2}}{2})^{0} (\frac{1}{x})^{6} : \frac{1}{x ^{6}}

= \frac{x ^{12}}{64} + \frac{3 x ^{9}}{16} + \frac{15 x ^{6}}{16} + \frac{5 x ^{3}}{2} + \frac{15}{4} + \frac{3}{x ^{3}} + \frac{1}{x ^{6}}

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