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Beliebt Algebra >

erweitern (2x+1/(3x))^{18}

Lösung

erweitern

(2 x + \frac{1}{3 x})^{18}

Lösung

262144 x^{18} + 786432 x^{16} + 1114112 x^{14} + \frac{8912896 x ^{12}}{9} + \frac{5570560 x ^{10}}{9} + \frac{7798784 x ^{8}}{27} + \frac{25346048 x ^{6}}{243} + \frac{7241728 x ^{4}}{243} + \frac{4978688 x ^{2}}{729} + \frac{24893440}{19683} + \frac{1244672}{6561 x ^{2}} + \frac{452608}{19683 x ^{4}} + \frac{396032}{177147 x ^{6}} + \frac{30464}{177147 x ^{8}} + \frac{5440}{531441 x ^{10}} + \frac{2176}{4782969 x ^{12}} + \frac{68}{4782969 x ^{14}} + \frac{4}{14348907 x ^{16}} + \frac{1}{387420489 x ^{18}}

Schritte zur Lösung

(2 x + \frac{1}{3 x})^{18}

Wende Binomialsetz an:

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

a = 2 x, b = \frac{1}{3 x}

= i = 0 \sum 18 (i 18) (2 x)^{(18 - i)} (\frac{1}{3 x})^{i}

Erweitere Summation

= \frac{18 !}{0 ! ( 18 - 0 ) !} (2 x)^{18} (\frac{1}{3 x})^{0} + \frac{18 !}{1 ! ( 18 - 1 ) !} (2 x)^{17} (\frac{1}{3 x})^{1} + \frac{18 !}{2 ! ( 18 - 2 ) !} (2 x)^{16} (\frac{1}{3 x})^{2} + \frac{18 !}{3 ! ( 18 - 3 ) !} (2 x)^{15} (\frac{1}{3 x})^{3} + \frac{18 !}{4 ! ( 18 - 4 ) !} (2 x)^{14} (\frac{1}{3 x})^{4} + \frac{18 !}{5 ! ( 18 - 5 ) !} (2 x)^{13} (\frac{1}{3 x})^{5} + \frac{18 !}{6 ! ( 18 - 6 ) !} (2 x)^{12} (\frac{1}{3 x})^{6} + \frac{18 !}{7 ! ( 18 - 7 ) !} (2 x)^{11} (\frac{1}{3 x})^{7} + \frac{18 !}{8 ! ( 18 - 8 ) !} (2 x)^{10} (\frac{1}{3 x})^{8} + \frac{18 !}{9 ! ( 18 - 9 ) !} (2 x)^{9} (\frac{1}{3 x})^{9} + \frac{18 !}{10 ! ( 18 - 10 ) !} (2 x)^{8} (\frac{1}{3 x})^{10} + \frac{18 !}{11 ! ( 18 - 11 ) !} (2 x)^{7} (\frac{1}{3 x})^{11} + \frac{18 !}{12 ! ( 18 - 12 ) !} (2 x)^{6} (\frac{1}{3 x})^{12} + \frac{18 !}{13 ! ( 18 - 13 ) !} (2 x)^{5} (\frac{1}{3 x})^{13} + \frac{18 !}{14 ! ( 18 - 14 ) !} (2 x)^{4} (\frac{1}{3 x})^{14} + \frac{18 !}{15 ! ( 18 - 15 ) !} (2 x)^{3} (\frac{1}{3 x})^{15} + \frac{18 !}{16 ! ( 18 - 16 ) !} (2 x)^{2} (\frac{1}{3 x})^{16} + \frac{18 !}{17 ! ( 18 - 17 ) !} (2 x)^{1} (\frac{1}{3 x})^{17} + \frac{18 !}{18 ! ( 18 - 18 ) !} (2 x)^{0} (\frac{1}{3 x})^{18}

Vereinfache

\frac{18 !}{0 ! ( 18 - 0 ) !} (2 x)^{18} (\frac{1}{3 x})^{0} : 262144 x^{18}

Vereinfache

\frac{18 !}{1 ! ( 18 - 1 ) !} (2 x)^{17} (\frac{1}{3 x})^{1} : 786432 x^{16}

Vereinfache

\frac{18 !}{2 ! ( 18 - 2 ) !} (2 x)^{16} (\frac{1}{3 x})^{2} : 1114112 x^{14}

Vereinfache

\frac{18 !}{3 ! ( 18 - 3 ) !} (2 x)^{15} (\frac{1}{3 x})^{3} : \frac{8912896 x ^{12}}{9}

Vereinfache

\frac{18 !}{4 ! ( 18 - 4 ) !} (2 x)^{14} (\frac{1}{3 x})^{4} : \frac{5570560 x ^{10}}{9}

Vereinfache

\frac{18 !}{5 ! ( 18 - 5 ) !} (2 x)^{13} (\frac{1}{3 x})^{5} : \frac{7798784 x ^{8}}{27}

Vereinfache

\frac{18 !}{6 ! ( 18 - 6 ) !} (2 x)^{12} (\frac{1}{3 x})^{6} : \frac{25346048 x ^{6}}{243}

Vereinfache

\frac{18 !}{7 ! ( 18 - 7 ) !} (2 x)^{11} (\frac{1}{3 x})^{7} : \frac{7241728 x ^{4}}{243}

Vereinfache

\frac{18 !}{8 ! ( 18 - 8 ) !} (2 x)^{10} (\frac{1}{3 x})^{8} : \frac{4978688 x ^{2}}{729}

Vereinfache

\frac{18 !}{9 ! ( 18 - 9 ) !} (2 x)^{9} (\frac{1}{3 x})^{9} : \frac{24893440}{19683}

Vereinfache

\frac{18 !}{10 ! ( 18 - 10 ) !} (2 x)^{8} (\frac{1}{3 x})^{10} : \frac{1244672}{6561 x ^{2}}

Vereinfache

\frac{18 !}{11 ! ( 18 - 11 ) !} (2 x)^{7} (\frac{1}{3 x})^{11} : \frac{452608}{19683 x ^{4}}

Vereinfache

\frac{18 !}{12 ! ( 18 - 12 ) !} (2 x)^{6} (\frac{1}{3 x})^{12} : \frac{396032}{177147 x ^{6}}

Vereinfache

\frac{18 !}{13 ! ( 18 - 13 ) !} (2 x)^{5} (\frac{1}{3 x})^{13} : \frac{30464}{177147 x ^{8}}

Vereinfache

\frac{18 !}{14 ! ( 18 - 14 ) !} (2 x)^{4} (\frac{1}{3 x})^{14} : \frac{5440}{531441 x ^{10}}

Vereinfache

\frac{18 !}{15 ! ( 18 - 15 ) !} (2 x)^{3} (\frac{1}{3 x})^{15} : \frac{2176}{4782969 x ^{12}}

Vereinfache

\frac{18 !}{16 ! ( 18 - 16 ) !} (2 x)^{2} (\frac{1}{3 x})^{16} : \frac{68}{4782969 x ^{14}}

Vereinfache

\frac{18 !}{17 ! ( 18 - 17 ) !} (2 x)^{1} (\frac{1}{3 x})^{17} : \frac{4}{14348907 x ^{16}}

Vereinfache

\frac{18 !}{18 ! ( 18 - 18 ) !} (2 x)^{0} (\frac{1}{3 x})^{18} : \frac{1}{387420489 x ^{18}}

= 262144 x^{18} + 786432 x^{16} + 1114112 x^{14} + \frac{8912896 x ^{12}}{9} + \frac{5570560 x ^{10}}{9} + \frac{7798784 x ^{8}}{27} + \frac{25346048 x ^{6}}{243} + \frac{7241728 x ^{4}}{243} + \frac{4978688 x ^{2}}{729} + \frac{24893440}{19683} + \frac{1244672}{6561 x ^{2}} + \frac{452608}{19683 x ^{4}} + \frac{396032}{177147 x ^{6}} + \frac{30464}{177147 x ^{8}} + \frac{5440}{531441 x ^{10}} + \frac{2176}{4782969 x ^{12}} + \frac{68}{4782969 x ^{14}} + \frac{4}{14348907 x ^{16}} + \frac{1}{387420489 x ^{18}}

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