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Beliebt Algebra >

erweitern (2x+y/4)^{16}

Lösung

erweitern

(2 x + \frac{y}{4})^{16}

Lösung

\frac{65536 \cdot 4 ^{16} x ^{16} + y ^{16}}{4 ^{16}} + 131072 x^{15} y + 122880 x^{14} y^{2} + 71680 x^{13} y^{3} + 29120 x^{12} y^{4} + 8736 x^{11} y^{5} + 2002 x^{10} y^{6} + \frac{715 x ^{9} y ^{7}}{2} + \frac{6435 x ^{8} y ^{8} + 715 x ^{7} y ^{9}}{128} + \frac{1001 x ^{6} y ^{10}}{2048} + \frac{273 x ^{5} y ^{11}}{8192} + \frac{455 x ^{4} y ^{12}}{262144} + \frac{35 x ^{3} y ^{13}}{524288} + \frac{15 x ^{2} y ^{14}}{8388608} + \frac{x y ^{15}}{33554432}

Schritte zur Lösung

(2 x + \frac{y}{4})^{16}

Wende Binomialsetz an:

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

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

= i = 0 \sum 16 (i 16) (2 x)^{(16 - i)} (\frac{y}{4})^{i}

Erweitere Summation

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

Vereinfache

\frac{16 !}{0 ! ( 16 - 0 ) !} (2 x)^{16} (\frac{y}{4})^{0} : 65536 x^{16}

Vereinfache

\frac{16 !}{1 ! ( 16 - 1 ) !} (2 x)^{15} (\frac{y}{4})^{1} : 131072 x^{15} y

Vereinfache

\frac{16 !}{2 ! ( 16 - 2 ) !} (2 x)^{14} (\frac{y}{4})^{2} : 122880 x^{14} y^{2}

Vereinfache

\frac{16 !}{3 ! ( 16 - 3 ) !} (2 x)^{13} (\frac{y}{4})^{3} : 71680 x^{13} y^{3}

Vereinfache

\frac{16 !}{4 ! ( 16 - 4 ) !} (2 x)^{12} (\frac{y}{4})^{4} : 29120 x^{12} y^{4}

Vereinfache

\frac{16 !}{5 ! ( 16 - 5 ) !} (2 x)^{11} (\frac{y}{4})^{5} : 8736 x^{11} y^{5}

Vereinfache

\frac{16 !}{6 ! ( 16 - 6 ) !} (2 x)^{10} (\frac{y}{4})^{6} : 2002 x^{10} y^{6}

Vereinfache

\frac{16 !}{7 ! ( 16 - 7 ) !} (2 x)^{9} (\frac{y}{4})^{7} : \frac{715 x ^{9} y ^{7}}{2}

Vereinfache

\frac{16 !}{8 ! ( 16 - 8 ) !} (2 x)^{8} (\frac{y}{4})^{8} : \frac{6435 x ^{8} y ^{8}}{128}

Vereinfache

\frac{16 !}{9 ! ( 16 - 9 ) !} (2 x)^{7} (\frac{y}{4})^{9} : \frac{715 x ^{7} y ^{9}}{128}

Vereinfache

\frac{16 !}{10 ! ( 16 - 10 ) !} (2 x)^{6} (\frac{y}{4})^{10} : \frac{1001 x ^{6} y ^{10}}{2048}

Vereinfache

\frac{16 !}{11 ! ( 16 - 11 ) !} (2 x)^{5} (\frac{y}{4})^{11} : \frac{273 x ^{5} y ^{11}}{8192}

Vereinfache

\frac{16 !}{12 ! ( 16 - 12 ) !} (2 x)^{4} (\frac{y}{4})^{12} : \frac{455 x ^{4} y ^{12}}{262144}

Vereinfache

\frac{16 !}{13 ! ( 16 - 13 ) !} (2 x)^{3} (\frac{y}{4})^{13} : \frac{35 x ^{3} y ^{13}}{524288}

Vereinfache

\frac{16 !}{14 ! ( 16 - 14 ) !} (2 x)^{2} (\frac{y}{4})^{14} : \frac{15 x ^{2} y ^{14}}{8388608}

Vereinfache

\frac{16 !}{15 ! ( 16 - 15 ) !} (2 x)^{1} (\frac{y}{4})^{15} : \frac{x y ^{15}}{33554432}

Vereinfache

\frac{16 !}{16 ! ( 16 - 16 ) !} (2 x)^{0} (\frac{y}{4})^{16} : \frac{y ^{16}}{4 ^{16}}

= 65536 x^{16} + 131072 x^{15} y + 122880 x^{14} y^{2} + 71680 x^{13} y^{3} + 29120 x^{12} y^{4} + 8736 x^{11} y^{5} + 2002 x^{10} y^{6} + \frac{715 x ^{9} y ^{7}}{2} + \frac{6435 x ^{8} y ^{8}}{128} + \frac{715 x ^{7} y ^{9}}{128} + \frac{1001 x ^{6} y ^{10}}{2048} + \frac{273 x ^{5} y ^{11}}{8192} + \frac{455 x ^{4} y ^{12}}{262144} + \frac{35 x ^{3} y ^{13}}{524288} + \frac{15 x ^{2} y ^{14}}{8388608} + \frac{x y ^{15}}{33554432} + \frac{y ^{16}}{4 ^{16}}

Vereinfache

65536 x^{16} + 131072 x^{15} y + 122880 x^{14} y^{2} + 71680 x^{13} y^{3} + 29120 x^{12} y^{4} + 8736 x^{11} y^{5} + 2002 x^{10} y^{6} + \frac{715 x ^{9} y ^{7}}{2} + \frac{6435 x ^{8} y ^{8}}{128} + \frac{715 x ^{7} y ^{9}}{128} + \frac{1001 x ^{6} y ^{10}}{2048} + \frac{273 x ^{5} y ^{11}}{8192} + \frac{455 x ^{4} y ^{12}}{262144} + \frac{35 x ^{3} y ^{13}}{524288} + \frac{15 x ^{2} y ^{14}}{8388608} + \frac{x y ^{15}}{33554432} + \frac{y ^{16}}{4 ^{16}} : \frac{65536 \cdot 4 ^{16} x ^{16} + y ^{16}}{4 ^{16}} + 131072 x^{15} y + 122880 x^{14} y^{2} + 71680 x^{13} y^{3} + 29120 x^{12} y^{4} + 8736 x^{11} y^{5} + 2002 x^{10} y^{6} + \frac{715 x ^{9} y ^{7}}{2} + \frac{6435 x ^{8} y ^{8} + 715 x ^{7} y ^{9}}{128} + \frac{1001 x ^{6} y ^{10}}{2048} + \frac{273 x ^{5} y ^{11}}{8192} + \frac{455 x ^{4} y ^{12}}{262144} + \frac{35 x ^{3} y ^{13}}{524288} + \frac{15 x ^{2} y ^{14}}{8388608} + \frac{x y ^{15}}{33554432}

= \frac{65536 \cdot 4 ^{16} x ^{16} + y ^{16}}{4 ^{16}} + 131072 x^{15} y + 122880 x^{14} y^{2} + 71680 x^{13} y^{3} + 29120 x^{12} y^{4} + 8736 x^{11} y^{5} + 2002 x^{10} y^{6} + \frac{715 x ^{9} y ^{7}}{2} + \frac{6435 x ^{8} y ^{8} + 715 x ^{7} y ^{9}}{128} + \frac{1001 x ^{6} y ^{10}}{2048} + \frac{273 x ^{5} y ^{11}}{8192} + \frac{455 x ^{4} y ^{12}}{262144} + \frac{35 x ^{3} y ^{13}}{524288} + \frac{15 x ^{2} y ^{14}}{8388608} + \frac{x y ^{15}}{33554432}

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