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Beliebt Algebra >

erweitern (1/4 x^2y+x/y)^{10}

Lösung

erweitern

(\frac{1}{4} x^{2} y + \frac{x}{y})^{10}

Lösung

\frac{x ^{20} y ^{10}}{1048576} + \frac{5 x ^{19} y ^{8}}{131072} + \frac{45 x ^{18} y ^{6}}{65536} + \frac{15 x ^{17} y ^{4} + 105 x ^{16} y ^{2}}{2048} + \frac{63 x ^{15}}{256} + \frac{105 x ^{14}}{128 y ^{2}} + \frac{15 x ^{13}}{8 y ^{4}} + \frac{45 x ^{12}}{16 y ^{6}} + \frac{5 x ^{11}}{2 y ^{8}} + \frac{x ^{10}}{y ^{10}}

Schritte zur Lösung

(\frac{1}{4} x^{2} y + \frac{x}{y})^{10}

Wende Binomialsetz an:

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

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

= i = 0 \sum 10 (i 10) (\frac{1}{4} x^{2} y)^{(10 - i)} (\frac{x}{y})^{i}

Erweitere Summation

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

Vereinfache

\frac{10 !}{0 ! ( 10 - 0 ) !} (\frac{1}{4} x^{2} y)^{10} (\frac{x}{y})^{0} : \frac{x ^{20} y ^{10}}{1048576}

Vereinfache

\frac{10 !}{1 ! ( 10 - 1 ) !} (\frac{1}{4} x^{2} y)^{9} (\frac{x}{y})^{1} : \frac{5 x ^{19} y ^{8}}{131072}

Vereinfache

\frac{10 !}{2 ! ( 10 - 2 ) !} (\frac{1}{4} x^{2} y)^{8} (\frac{x}{y})^{2} : \frac{45 x ^{18} y ^{6}}{65536}

Vereinfache

\frac{10 !}{3 ! ( 10 - 3 ) !} (\frac{1}{4} x^{2} y)^{7} (\frac{x}{y})^{3} : \frac{15 x ^{17} y ^{4}}{2048}

Vereinfache

\frac{10 !}{4 ! ( 10 - 4 ) !} (\frac{1}{4} x^{2} y)^{6} (\frac{x}{y})^{4} : \frac{105 x ^{16} y ^{2}}{2048}

Vereinfache

\frac{10 !}{5 ! ( 10 - 5 ) !} (\frac{1}{4} x^{2} y)^{5} (\frac{x}{y})^{5} : \frac{63 x ^{15}}{256}

Vereinfache

\frac{10 !}{6 ! ( 10 - 6 ) !} (\frac{1}{4} x^{2} y)^{4} (\frac{x}{y})^{6} : \frac{105 x ^{14}}{128 y ^{2}}

Vereinfache

\frac{10 !}{7 ! ( 10 - 7 ) !} (\frac{1}{4} x^{2} y)^{3} (\frac{x}{y})^{7} : \frac{15 x ^{13}}{8 y ^{4}}

Vereinfache

\frac{10 !}{8 ! ( 10 - 8 ) !} (\frac{1}{4} x^{2} y)^{2} (\frac{x}{y})^{8} : \frac{45 x ^{12}}{16 y ^{6}}

Vereinfache

\frac{10 !}{9 ! ( 10 - 9 ) !} (\frac{1}{4} x^{2} y)^{1} (\frac{x}{y})^{9} : \frac{5 x ^{11}}{2 y ^{8}}

Vereinfache

\frac{10 !}{10 ! ( 10 - 10 ) !} (\frac{1}{4} x^{2} y)^{0} (\frac{x}{y})^{10} : \frac{x ^{10}}{y ^{10}}

= \frac{x ^{20} y ^{10}}{1048576} + \frac{5 x ^{19} y ^{8}}{131072} + \frac{45 x ^{18} y ^{6}}{65536} + \frac{15 x ^{17} y ^{4}}{2048} + \frac{105 x ^{16} y ^{2}}{2048} + \frac{63 x ^{15}}{256} + \frac{105 x ^{14}}{128 y ^{2}} + \frac{15 x ^{13}}{8 y ^{4}} + \frac{45 x ^{12}}{16 y ^{6}} + \frac{5 x ^{11}}{2 y ^{8}} + \frac{x ^{10}}{y ^{10}}

Füge zusammen:

\frac{x ^{20} y ^{10}}{1048576} + \frac{5 x ^{19} y ^{8}}{131072} + \frac{45 x ^{18} y ^{6}}{65536} + \frac{15 x ^{17} y ^{4}}{2048} + \frac{105 x ^{16} y ^{2}}{2048} + \frac{63 x ^{15}}{256} + \frac{105 x ^{14}}{128 y ^{2}} + \frac{15 x ^{13}}{8 y ^{4}} + \frac{45 x ^{12}}{16 y ^{6}} + \frac{5 x ^{11}}{2 y ^{8}} + \frac{x ^{10}}{y ^{10}} : \frac{x ^{20} y ^{10}}{1048576} + \frac{5 x ^{19} y ^{8}}{131072} + \frac{45 x ^{18} y ^{6}}{65536} + \frac{15 x ^{17} y ^{4} + 105 x ^{16} y ^{2}}{2048} + \frac{63 x ^{15}}{256} + \frac{105 x ^{14}}{128 y ^{2}} + \frac{15 x ^{13}}{8 y ^{4}} + \frac{45 x ^{12}}{16 y ^{6}} + \frac{5 x ^{11}}{2 y ^{8}} + \frac{x ^{10}}{y ^{10}}

= \frac{x ^{20} y ^{10}}{1048576} + \frac{5 x ^{19} y ^{8}}{131072} + \frac{45 x ^{18} y ^{6}}{65536} + \frac{15 x ^{17} y ^{4} + 105 x ^{16} y ^{2}}{2048} + \frac{63 x ^{15}}{256} + \frac{105 x ^{14}}{128 y ^{2}} + \frac{15 x ^{13}}{8 y ^{4}} + \frac{45 x ^{12}}{16 y ^{6}} + \frac{5 x ^{11}}{2 y ^{8}} + \frac{x ^{10}}{y ^{10}}

Beliebte Beispiele

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