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Beliebt Algebra >

erweitern (2x+1/(4x))^{20}

Lösung

erweitern

(2 x + \frac{1}{4 x})^{20}

Lösung

1048576 x^{20} + 2621440 x^{18} + 3112960 x^{16} + 2334720 x^{14} + 1240320 x^{12} + 496128 x^{10} + 155040 x^{8} + 38760 x^{6} + \frac{62985 x ^{4}}{8} + \frac{20995 x ^{2}}{16} + \frac{46189}{256} + \frac{20995}{1024 x ^{2}} + \frac{62985}{32768 x ^{4}} + \frac{4845}{32768 x ^{6}} + \frac{4845}{524288 x ^{8}} + \frac{969}{2097152 x ^{10}} + \frac{4845}{268435456 x ^{12}} + \frac{285}{536870912 x ^{14}} + \frac{95}{2 ^{33} x ^{16}} + \frac{5}{2 ^{35} x ^{18}} + \frac{1}{4 ^{20} x ^{20}}

Schritte zur Lösung

(2 x + \frac{1}{4 x})^{20}

Wende Binomialsetz an:

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

a = 2 x, b = \frac{1}{4 x}

= i = 0 \sum 20 (i 20) (2 x)^{(20 - i)} (\frac{1}{4 x})^{i}

Erweitere Summation

= \frac{20 !}{0 ! ( 20 - 0 ) !} (2 x)^{20} (\frac{1}{4 x})^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} (2 x)^{19} (\frac{1}{4 x})^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} (2 x)^{18} (\frac{1}{4 x})^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} (2 x)^{17} (\frac{1}{4 x})^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} (2 x)^{16} (\frac{1}{4 x})^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} (2 x)^{15} (\frac{1}{4 x})^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} (2 x)^{14} (\frac{1}{4 x})^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} (2 x)^{13} (\frac{1}{4 x})^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} (2 x)^{12} (\frac{1}{4 x})^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} (2 x)^{11} (\frac{1}{4 x})^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} (2 x)^{10} (\frac{1}{4 x})^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} (2 x)^{9} (\frac{1}{4 x})^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} (2 x)^{8} (\frac{1}{4 x})^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} (2 x)^{7} (\frac{1}{4 x})^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} (2 x)^{6} (\frac{1}{4 x})^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} (2 x)^{5} (\frac{1}{4 x})^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} (2 x)^{4} (\frac{1}{4 x})^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} (2 x)^{3} (\frac{1}{4 x})^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} (2 x)^{2} (\frac{1}{4 x})^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} (2 x)^{1} (\frac{1}{4 x})^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} (2 x)^{0} (\frac{1}{4 x})^{20}

Vereinfache

\frac{20 !}{0 ! ( 20 - 0 ) !} (2 x)^{20} (\frac{1}{4 x})^{0} : 1048576 x^{20}

Vereinfache

\frac{20 !}{1 ! ( 20 - 1 ) !} (2 x)^{19} (\frac{1}{4 x})^{1} : 2621440 x^{18}

Vereinfache

\frac{20 !}{2 ! ( 20 - 2 ) !} (2 x)^{18} (\frac{1}{4 x})^{2} : 3112960 x^{16}

Vereinfache

\frac{20 !}{3 ! ( 20 - 3 ) !} (2 x)^{17} (\frac{1}{4 x})^{3} : 2334720 x^{14}

Vereinfache

\frac{20 !}{4 ! ( 20 - 4 ) !} (2 x)^{16} (\frac{1}{4 x})^{4} : 1240320 x^{12}

Vereinfache

\frac{20 !}{5 ! ( 20 - 5 ) !} (2 x)^{15} (\frac{1}{4 x})^{5} : 496128 x^{10}

Vereinfache

\frac{20 !}{6 ! ( 20 - 6 ) !} (2 x)^{14} (\frac{1}{4 x})^{6} : 155040 x^{8}

Vereinfache

\frac{20 !}{7 ! ( 20 - 7 ) !} (2 x)^{13} (\frac{1}{4 x})^{7} : 38760 x^{6}

Vereinfache

\frac{20 !}{8 ! ( 20 - 8 ) !} (2 x)^{12} (\frac{1}{4 x})^{8} : \frac{62985 x ^{4}}{8}

Vereinfache

\frac{20 !}{9 ! ( 20 - 9 ) !} (2 x)^{11} (\frac{1}{4 x})^{9} : \frac{20995 x ^{2}}{16}

Vereinfache

\frac{20 !}{10 ! ( 20 - 10 ) !} (2 x)^{10} (\frac{1}{4 x})^{10} : \frac{46189}{256}

Vereinfache

\frac{20 !}{11 ! ( 20 - 11 ) !} (2 x)^{9} (\frac{1}{4 x})^{11} : \frac{20995}{1024 x ^{2}}

Vereinfache

\frac{20 !}{12 ! ( 20 - 12 ) !} (2 x)^{8} (\frac{1}{4 x})^{12} : \frac{62985}{32768 x ^{4}}

Vereinfache

\frac{20 !}{13 ! ( 20 - 13 ) !} (2 x)^{7} (\frac{1}{4 x})^{13} : \frac{4845}{32768 x ^{6}}

Vereinfache

\frac{20 !}{14 ! ( 20 - 14 ) !} (2 x)^{6} (\frac{1}{4 x})^{14} : \frac{4845}{524288 x ^{8}}

Vereinfache

\frac{20 !}{15 ! ( 20 - 15 ) !} (2 x)^{5} (\frac{1}{4 x})^{15} : \frac{969}{2097152 x ^{10}}

Vereinfache

\frac{20 !}{16 ! ( 20 - 16 ) !} (2 x)^{4} (\frac{1}{4 x})^{16} : \frac{4845}{268435456 x ^{12}}

Vereinfache

\frac{20 !}{17 ! ( 20 - 17 ) !} (2 x)^{3} (\frac{1}{4 x})^{17} : \frac{285}{536870912 x ^{14}}

Vereinfache

\frac{20 !}{18 ! ( 20 - 18 ) !} (2 x)^{2} (\frac{1}{4 x})^{18} : \frac{95}{2 ^{33} x ^{16}}

Vereinfache

\frac{20 !}{19 ! ( 20 - 19 ) !} (2 x)^{1} (\frac{1}{4 x})^{19} : \frac{5}{2 ^{35} x ^{18}}

Vereinfache

\frac{20 !}{20 ! ( 20 - 20 ) !} (2 x)^{0} (\frac{1}{4 x})^{20} : \frac{1}{4 ^{20} x ^{20}}

= 1048576 x^{20} + 2621440 x^{18} + 3112960 x^{16} + 2334720 x^{14} + 1240320 x^{12} + 496128 x^{10} + 155040 x^{8} + 38760 x^{6} + \frac{62985 x ^{4}}{8} + \frac{20995 x ^{2}}{16} + \frac{46189}{256} + \frac{20995}{1024 x ^{2}} + \frac{62985}{32768 x ^{4}} + \frac{4845}{32768 x ^{6}} + \frac{4845}{524288 x ^{8}} + \frac{969}{2097152 x ^{10}} + \frac{4845}{268435456 x ^{12}} + \frac{285}{536870912 x ^{14}} + \frac{95}{2 ^{33} x ^{16}} + \frac{5}{2 ^{35} x ^{18}} + \frac{1}{4 ^{20} x ^{20}}

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