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Beliebt Algebra >

erweitern (x^2+2/x)^9

Lösung

erweitern

(x^{2} + \frac{2}{x})^{9}

Lösung

x^{18} + 18 x^{15} + 144 x^{12} + 672 x^{9} + 2016 x^{6} + 4032 x^{3} + 5376 + \frac{4608}{x ^{3}} + \frac{2304}{x ^{6}} + \frac{512}{x ^{9}}

Schritte zur Lösung

(x^{2} + \frac{2}{x})^{9}

Wende Binomialsetz an:

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

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

= i = 0 \sum 9 (i 9) (x^{2})^{(9 - i)} (\frac{2}{x})^{i}

Erweitere Summation

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

Vereinfache

\frac{9 !}{0 ! ( 9 - 0 ) !} (x^{2})^{9} (\frac{2}{x})^{0} : x^{18}

Vereinfache

\frac{9 !}{1 ! ( 9 - 1 ) !} (x^{2})^{8} (\frac{2}{x})^{1} : 18 x^{15}

Vereinfache

\frac{9 !}{2 ! ( 9 - 2 ) !} (x^{2})^{7} (\frac{2}{x})^{2} : 144 x^{12}

Vereinfache

\frac{9 !}{3 ! ( 9 - 3 ) !} (x^{2})^{6} (\frac{2}{x})^{3} : 672 x^{9}

Vereinfache

\frac{9 !}{4 ! ( 9 - 4 ) !} (x^{2})^{5} (\frac{2}{x})^{4} : 2016 x^{6}

Vereinfache

\frac{9 !}{5 ! ( 9 - 5 ) !} (x^{2})^{4} (\frac{2}{x})^{5} : 4032 x^{3}

Vereinfache

\frac{9 !}{6 ! ( 9 - 6 ) !} (x^{2})^{3} (\frac{2}{x})^{6} : 5376

Vereinfache

\frac{9 !}{7 ! ( 9 - 7 ) !} (x^{2})^{2} (\frac{2}{x})^{7} : \frac{4608}{x ^{3}}

Vereinfache

\frac{9 !}{8 ! ( 9 - 8 ) !} (x^{2})^{1} (\frac{2}{x})^{8} : \frac{2304}{x ^{6}}

Vereinfache

\frac{9 !}{9 ! ( 9 - 9 ) !} (x^{2})^{0} (\frac{2}{x})^{9} : \frac{512}{x ^{9}}

= x^{18} + 18 x^{15} + 144 x^{12} + 672 x^{9} + 2016 x^{6} + 4032 x^{3} + 5376 + \frac{4608}{x ^{3}} + \frac{2304}{x ^{6}} + \frac{512}{x ^{9}}

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