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Populaire Algèbre >

développer (x^2+4/x)^{12}

Solution

développer

(x^{2} + \frac{4}{x})^{12}

Solution

x^{24} + 48 x^{21} + 1056 x^{18} + 14080 x^{15} + 126720 x^{12} + 811008 x^{9} + 3784704 x^{6} + 12976128 x^{3} + 32440320 + \frac{57671680}{x ^{3}} + \frac{69206016}{x ^{6}} + \frac{50331648}{x ^{9}} + \frac{16777216}{x ^{12}}

étapes des solutions

(x^{2} + \frac{4}{x})^{12}

Appliquer le théorème du binôme:

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

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

= i = 0 \sum 12 (i 12) (x^{2})^{(12 - i)} (\frac{4}{x})^{i}

Développer la somme

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

Simplifier

\frac{12 !}{0 ! ( 12 - 0 ) !} (x^{2})^{12} (\frac{4}{x})^{0} : x^{24}

Simplifier

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{2})^{11} (\frac{4}{x})^{1} : 48 x^{21}

Simplifier

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{2})^{10} (\frac{4}{x})^{2} : 1056 x^{18}

Simplifier

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{2})^{9} (\frac{4}{x})^{3} : 14080 x^{15}

Simplifier

\frac{12 !}{4 ! ( 12 - 4 ) !} (x^{2})^{8} (\frac{4}{x})^{4} : 126720 x^{12}

Simplifier

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{2})^{7} (\frac{4}{x})^{5} : 811008 x^{9}

Simplifier

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{2})^{6} (\frac{4}{x})^{6} : 3784704 x^{6}

Simplifier

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{2})^{5} (\frac{4}{x})^{7} : 12976128 x^{3}

Simplifier

\frac{12 !}{8 ! ( 12 - 8 ) !} (x^{2})^{4} (\frac{4}{x})^{8} : 32440320

Simplifier

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{2})^{3} (\frac{4}{x})^{9} : \frac{57671680}{x ^{3}}

Simplifier

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{2})^{2} (\frac{4}{x})^{10} : \frac{69206016}{x ^{6}}

Simplifier

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{2})^{1} (\frac{4}{x})^{11} : \frac{50331648}{x ^{9}}

Simplifier

\frac{12 !}{12 ! ( 12 - 12 ) !} (x^{2})^{0} (\frac{4}{x})^{12} : \frac{16777216}{x ^{12}}

= x^{24} + 48 x^{21} + 1056 x^{18} + 14080 x^{15} + 126720 x^{12} + 811008 x^{9} + 3784704 x^{6} + 12976128 x^{3} + 32440320 + \frac{57671680}{x ^{3}} + \frac{69206016}{x ^{6}} + \frac{50331648}{x ^{9}} + \frac{16777216}{x ^{12}}

Exemples populaires

développer 2/((s-1)(s^2+1))

e x p an d \frac{2}{( s - 1 ) ( s ^{2} + 1 )}

développer-5r^2sin(θ)(r^2-9)

e x p an d - 5 r^{2} sin (θ) (r^{2} - 9)

développer 3/(n(n+3))

e x p an d \frac{3}{n ( n + 3 )}

développer (2x+1)/(x(x+1))

e x p an d \frac{2 x + 1}{x ( x + 1 )}

développer sqrt((x-1)^2+y^2)

e x p an d (x - 1)^{2} + y^{2}