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Beliebt Algebra >

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

Lösung

erweitern

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

Lösung

x^{24} + 36 x^{21} + 594 x^{18} + 5940 x^{15} + 40095 x^{12} + 192456 x^{9} + 673596 x^{6} + 1732104 x^{3} + 3247695 + \frac{4330260}{x ^{3}} + \frac{3897234}{x ^{6}} + \frac{2125764}{x ^{9}} + \frac{531441}{x ^{12}}

Schritte zur Lösung

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

Wende Binomialsetz an:

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

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

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

Erweitere Summation

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

Vereinfache

\frac{12 !}{0 ! ( 12 - 0 ) !} (x^{2})^{12} (\frac{3}{x})^{0} : x^{24}

Vereinfache

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{2})^{11} (\frac{3}{x})^{1} : 36 x^{21}

Vereinfache

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{2})^{10} (\frac{3}{x})^{2} : 594 x^{18}

Vereinfache

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{2})^{9} (\frac{3}{x})^{3} : 5940 x^{15}

Vereinfache

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

Vereinfache

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{2})^{7} (\frac{3}{x})^{5} : 192456 x^{9}

Vereinfache

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{2})^{6} (\frac{3}{x})^{6} : 673596 x^{6}

Vereinfache

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{2})^{5} (\frac{3}{x})^{7} : 1732104 x^{3}

Vereinfache

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

Vereinfache

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{2})^{3} (\frac{3}{x})^{9} : \frac{4330260}{x ^{3}}

Vereinfache

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{2})^{2} (\frac{3}{x})^{10} : \frac{3897234}{x ^{6}}

Vereinfache

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{2})^{1} (\frac{3}{x})^{11} : \frac{2125764}{x ^{9}}

Vereinfache

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

= x^{24} + 36 x^{21} + 594 x^{18} + 5940 x^{15} + 40095 x^{12} + 192456 x^{9} + 673596 x^{6} + 1732104 x^{3} + 3247695 + \frac{4330260}{x ^{3}} + \frac{3897234}{x ^{6}} + \frac{2125764}{x ^{9}} + \frac{531441}{x ^{12}}

Beliebte Beispiele

erweitern (24xy^3-3e^xy-3)/(e^xy^2-8xy^4+y)

e x p an d \frac{24 x y ^{3} - 3 e ^{x} y - 3}{e ^{x} y ^{2} - 8 x y ^{4} + y}

erweitern (-3,-3,6)y(4,8,-5)

e x p an d (- 3, - 3, 6) y (4, 8, - 5)

erweitern B(3,2)yC(-3,-2)

e x p an d B (3, 2) y C (- 3, - 2)

erweitern 1/4 (36a+b)

e x p an d \frac{1}{4} (36 a + b)

erweitern 4(x-3/4 = 5/6)

e x p an d 4 (x - \frac{3}{4} = \frac{5}{6})