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Beliebt Algebra >

erweitern (x+2/(x^2))^{15}

Lösung

erweitern

(x + \frac{2}{x ^{2}})^{15}

Lösung

x^{15} + 30 x^{12} + 420 x^{9} + 3640 x^{6} + 21840 x^{3} + 96096 + \frac{320320}{x ^{3}} + \frac{823680}{x ^{6}} + \frac{1647360}{x ^{9}} + \frac{2562560}{x ^{12}} + \frac{3075072}{x ^{15}} + \frac{2795520}{x ^{18}} + \frac{1863680}{x ^{21}} + \frac{860160}{x ^{24}} + \frac{245760}{x ^{27}} + \frac{32768}{x ^{30}}

Schritte zur Lösung

(x + \frac{2}{x ^{2}})^{15}

Wende Binomialsetz an:

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

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

= i = 0 \sum 15 (i 15) x^{(15 - i)} (\frac{2}{x ^{2}})^{i}

Erweitere Summation

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

Vereinfache

\frac{15 !}{0 ! ( 15 - 0 ) !} x^{15} (\frac{2}{x ^{2}})^{0} : x^{15}

Vereinfache

\frac{15 !}{1 ! ( 15 - 1 ) !} x^{14} (\frac{2}{x ^{2}})^{1} : 30 x^{12}

Vereinfache

\frac{15 !}{2 ! ( 15 - 2 ) !} x^{13} (\frac{2}{x ^{2}})^{2} : 420 x^{9}

Vereinfache

\frac{15 !}{3 ! ( 15 - 3 ) !} x^{12} (\frac{2}{x ^{2}})^{3} : 3640 x^{6}

Vereinfache

\frac{15 !}{4 ! ( 15 - 4 ) !} x^{11} (\frac{2}{x ^{2}})^{4} : 21840 x^{3}

Vereinfache

\frac{15 !}{5 ! ( 15 - 5 ) !} x^{10} (\frac{2}{x ^{2}})^{5} : 96096

Vereinfache

\frac{15 !}{6 ! ( 15 - 6 ) !} x^{9} (\frac{2}{x ^{2}})^{6} : \frac{320320}{x ^{3}}

Vereinfache

\frac{15 !}{7 ! ( 15 - 7 ) !} x^{8} (\frac{2}{x ^{2}})^{7} : \frac{823680}{x ^{6}}

Vereinfache

\frac{15 !}{8 ! ( 15 - 8 ) !} x^{7} (\frac{2}{x ^{2}})^{8} : \frac{1647360}{x ^{9}}

Vereinfache

\frac{15 !}{9 ! ( 15 - 9 ) !} x^{6} (\frac{2}{x ^{2}})^{9} : \frac{2562560}{x ^{12}}

Vereinfache

\frac{15 !}{10 ! ( 15 - 10 ) !} x^{5} (\frac{2}{x ^{2}})^{10} : \frac{3075072}{x ^{15}}

Vereinfache

\frac{15 !}{11 ! ( 15 - 11 ) !} x^{4} (\frac{2}{x ^{2}})^{11} : \frac{2795520}{x ^{18}}

Vereinfache

\frac{15 !}{12 ! ( 15 - 12 ) !} x^{3} (\frac{2}{x ^{2}})^{12} : \frac{1863680}{x ^{21}}

Vereinfache

\frac{15 !}{13 ! ( 15 - 13 ) !} x^{2} (\frac{2}{x ^{2}})^{13} : \frac{860160}{x ^{24}}

Vereinfache

\frac{15 !}{14 ! ( 15 - 14 ) !} x^{1} (\frac{2}{x ^{2}})^{14} : \frac{245760}{x ^{27}}

Vereinfache

\frac{15 !}{15 ! ( 15 - 15 ) !} x^{0} (\frac{2}{x ^{2}})^{15} : \frac{32768}{x ^{30}}

= x^{15} + 30 x^{12} + 420 x^{9} + 3640 x^{6} + 21840 x^{3} + 96096 + \frac{320320}{x ^{3}} + \frac{823680}{x ^{6}} + \frac{1647360}{x ^{9}} + \frac{2562560}{x ^{12}} + \frac{3075072}{x ^{15}} + \frac{2795520}{x ^{18}} + \frac{1863680}{x ^{21}} + \frac{860160}{x ^{24}} + \frac{245760}{x ^{27}} + \frac{32768}{x ^{30}}

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