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Beliebt Algebra >

erweitern (x+3/2)^{12}

Lösung

erweitern

(x + \frac{3}{2})^{12}

Lösung

x^{12} + 18 x^{11} + \frac{297 x ^{10}}{2} + \frac{1485 x ^{9}}{2} + \frac{40095 x ^{8}}{16} + \frac{24057 x ^{7}}{4} + \frac{168399 x ^{6}}{16} + \frac{216513 x ^{5}}{16} + \frac{3247695 x ^{4}}{256} + \frac{1082565 x ^{3}}{128} + \frac{1948617 x ^{2}}{512} + \frac{531441 x}{512} + \frac{531441}{4096}

Schritte zur Lösung

(x + \frac{3}{2})^{12}

Wende Binomialsetz an:

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

a = x, b = \frac{3}{2}

= i = 0 \sum 12 (i 12) x^{(12 - i)} (\frac{3}{2})^{i}

Erweitere Summation

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

Vereinfache

\frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} (\frac{3}{2})^{0} : x^{12}

Vereinfache

\frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} (\frac{3}{2})^{1} : 18 x^{11}

Vereinfache

\frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} (\frac{3}{2})^{2} : \frac{297 x ^{10}}{2}

Vereinfache

\frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} (\frac{3}{2})^{3} : \frac{1485 x ^{9}}{2}

Vereinfache

\frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} (\frac{3}{2})^{4} : \frac{40095 x ^{8}}{16}

Vereinfache

\frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} (\frac{3}{2})^{5} : \frac{24057 x ^{7}}{4}

Vereinfache

\frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} (\frac{3}{2})^{6} : \frac{168399 x ^{6}}{16}

Vereinfache

\frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} (\frac{3}{2})^{7} : \frac{216513 x ^{5}}{16}

Vereinfache

\frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} (\frac{3}{2})^{8} : \frac{3247695 x ^{4}}{256}

Vereinfache

\frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} (\frac{3}{2})^{9} : \frac{1082565 x ^{3}}{128}

Vereinfache

\frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} (\frac{3}{2})^{10} : \frac{1948617 x ^{2}}{512}

Vereinfache

\frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} (\frac{3}{2})^{11} : \frac{531441 x}{512}

Vereinfache

\frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} (\frac{3}{2})^{12} : \frac{531441}{4096}

= x^{12} + 18 x^{11} + \frac{297 x ^{10}}{2} + \frac{1485 x ^{9}}{2} + \frac{40095 x ^{8}}{16} + \frac{24057 x ^{7}}{4} + \frac{168399 x ^{6}}{16} + \frac{216513 x ^{5}}{16} + \frac{3247695 x ^{4}}{256} + \frac{1082565 x ^{3}}{128} + \frac{1948617 x ^{2}}{512} + \frac{531441 x}{512} + \frac{531441}{4096}

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