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Beliebt Algebra >

erweitern (x^2+2/x)^{10}

Lösung

erweitern

(x^{2} + \frac{2}{x})^{10}

Lösung

x^{20} + 20 x^{17} + 180 x^{14} + 960 x^{11} + 3360 x^{8} + 8064 x^{5} + 13440 x^{2} + \frac{15360}{x} + \frac{11520}{x ^{4}} + \frac{5120}{x ^{7}} + \frac{1024}{x ^{10}}

Schritte zur Lösung

(x^{2} + \frac{2}{x})^{10}

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 10 (i 10) (x^{2})^{(10 - i)} (\frac{2}{x})^{i}

Erweitere Summation

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

Vereinfache

\frac{10 !}{0 ! ( 10 - 0 ) !} (x^{2})^{10} (\frac{2}{x})^{0} : x^{20}

Vereinfache

\frac{10 !}{1 ! ( 10 - 1 ) !} (x^{2})^{9} (\frac{2}{x})^{1} : 20 x^{17}

Vereinfache

\frac{10 !}{2 ! ( 10 - 2 ) !} (x^{2})^{8} (\frac{2}{x})^{2} : 180 x^{14}

Vereinfache

\frac{10 !}{3 ! ( 10 - 3 ) !} (x^{2})^{7} (\frac{2}{x})^{3} : 960 x^{11}

Vereinfache

\frac{10 !}{4 ! ( 10 - 4 ) !} (x^{2})^{6} (\frac{2}{x})^{4} : 3360 x^{8}

Vereinfache

\frac{10 !}{5 ! ( 10 - 5 ) !} (x^{2})^{5} (\frac{2}{x})^{5} : 8064 x^{5}

Vereinfache

\frac{10 !}{6 ! ( 10 - 6 ) !} (x^{2})^{4} (\frac{2}{x})^{6} : 13440 x^{2}

Vereinfache

\frac{10 !}{7 ! ( 10 - 7 ) !} (x^{2})^{3} (\frac{2}{x})^{7} : \frac{15360}{x}

Vereinfache

\frac{10 !}{8 ! ( 10 - 8 ) !} (x^{2})^{2} (\frac{2}{x})^{8} : \frac{11520}{x ^{4}}

Vereinfache

\frac{10 !}{9 ! ( 10 - 9 ) !} (x^{2})^{1} (\frac{2}{x})^{9} : \frac{5120}{x ^{7}}

Vereinfache

\frac{10 !}{10 ! ( 10 - 10 ) !} (x^{2})^{0} (\frac{2}{x})^{10} : \frac{1024}{x ^{10}}

= x^{20} + 20 x^{17} + 180 x^{14} + 960 x^{11} + 3360 x^{8} + 8064 x^{5} + 13440 x^{2} + \frac{15360}{x} + \frac{11520}{x ^{4}} + \frac{5120}{x ^{7}} + \frac{1024}{x ^{10}}

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