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Beliebt Algebra >

erweitern (3x+(4/5)/x)^8

Lösung

erweitern

(3 x + \frac{\frac{4}{5}}{x})^{8}

Lösung

6561 x^{8} + \frac{69984 x ^{6}}{5} + \frac{326592 x ^{4}}{25} + \frac{870912 x ^{2}}{125} + \frac{290304}{125} + \frac{1548288}{3125 x ^{2}} + \frac{1032192}{15625 x ^{4}} + \frac{393216}{78125 x ^{6}} + \frac{65536}{390625 x ^{8}}

Schritte zur Lösung

(3 x + \frac{\frac{4}{5}}{x})^{8}

Wende Binomialsetz an:

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

a = 3 x, b = \frac{\frac{4}{5}}{x}

= i = 0 \sum 8 (i 8) (3 x)^{(8 - i)} (\frac{\frac{4}{5}}{x})^{i}

Erweitere Summation

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

Vereinfache

\frac{8 !}{0 ! ( 8 - 0 ) !} (3 x)^{8} (\frac{\frac{4}{5}}{x})^{0} : 6561 x^{8}

Vereinfache

\frac{8 !}{1 ! ( 8 - 1 ) !} (3 x)^{7} (\frac{\frac{4}{5}}{x})^{1} : \frac{69984 x ^{6}}{5}

Vereinfache

\frac{8 !}{2 ! ( 8 - 2 ) !} (3 x)^{6} (\frac{\frac{4}{5}}{x})^{2} : \frac{326592 x ^{4}}{25}

Vereinfache

\frac{8 !}{3 ! ( 8 - 3 ) !} (3 x)^{5} (\frac{\frac{4}{5}}{x})^{3} : \frac{870912 x ^{2}}{125}

Vereinfache

\frac{8 !}{4 ! ( 8 - 4 ) !} (3 x)^{4} (\frac{\frac{4}{5}}{x})^{4} : \frac{290304}{125}

Vereinfache

\frac{8 !}{5 ! ( 8 - 5 ) !} (3 x)^{3} (\frac{\frac{4}{5}}{x})^{5} : \frac{1548288}{3125 x ^{2}}

Vereinfache

\frac{8 !}{6 ! ( 8 - 6 ) !} (3 x)^{2} (\frac{\frac{4}{5}}{x})^{6} : \frac{1032192}{15625 x ^{4}}

Vereinfache

\frac{8 !}{7 ! ( 8 - 7 ) !} (3 x)^{1} (\frac{\frac{4}{5}}{x})^{7} : \frac{393216}{78125 x ^{6}}

Vereinfache

\frac{8 !}{8 ! ( 8 - 8 ) !} (3 x)^{0} (\frac{\frac{4}{5}}{x})^{8} : \frac{65536}{390625 x ^{8}}

= 6561 x^{8} + \frac{69984 x ^{6}}{5} + \frac{326592 x ^{4}}{25} + \frac{870912 x ^{2}}{125} + \frac{290304}{125} + \frac{1548288}{3125 x ^{2}} + \frac{1032192}{15625 x ^{4}} + \frac{393216}{78125 x ^{6}} + \frac{65536}{390625 x ^{8}}

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