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Populaire Algèbre >

développer (81x+6)^{26}

Solution

développer

(81 x + 6)^{26}

Solution

8 1^{26} x^{26} + 8 1^{25} \cdot 156 x^{25} + 3^{98} \cdot 1300 x^{24} + 3^{95} \cdot 20800 x^{23} + 3^{92} \cdot 239200 x^{22} + 3^{89} \cdot 2104960 x^{21} + 3^{86} \cdot 14734720 x^{20} + 3^{83} \cdot 84198400 x^{19} + 3^{80} \cdot 399942400 x^{18} + 3^{77} \cdot 1599769600 x^{17} + 3^{74} \cdot 5439216640 x^{16} + 3^{71} \cdot 15823175680 x^{15} + 3^{68} \cdot 39557939200 x^{14} + \frac{3 ^{60} \cdot 7.57933 E 19 x ^{13}}{3660800} + \frac{3 ^{57} \cdot 1.08276 E 19 x ^{12}}{281600} + 3^{55} \cdot 2.05068 E 13 x^{11} + 3^{56} \cdot 348109864960 x^{10} + 3^{53} \cdot 409541017600 x^{9} + 3^{50} \cdot 409541017600 x^{8} + 3^{47} \cdot 344876646400 x^{7} + 3^{44} \cdot 241413652480 x^{6} + 3^{41} \cdot 137950658560 x^{5} + 3^{38} \cdot 62704844800 x^{4} + 3^{35} \cdot 21810380800 x^{3} + 3^{32} \cdot 5452595200 x^{2} + 6^{25} \cdot 2106 x + 6^{26}

étapes des solutions

(81 x + 6)^{26}

Appliquer le théorème du binôme:

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

a = 81 x, b = 6

= i = 0 \sum 26 (i 26) (81 x)^{(26 - i)} \cdot 6^{i}

Développer la somme

= \frac{26 !}{0 ! ( 26 - 0 ) !} (81 x)^{26} \cdot 6^{0} + \frac{26 !}{1 ! ( 26 - 1 ) !} (81 x)^{25} \cdot 6^{1} + \frac{26 !}{2 ! ( 26 - 2 ) !} (81 x)^{24} \cdot 6^{2} + \frac{26 !}{3 ! ( 26 - 3 ) !} (81 x)^{23} \cdot 6^{3} + \frac{26 !}{4 ! ( 26 - 4 ) !} (81 x)^{22} \cdot 6^{4} + \frac{26 !}{5 ! ( 26 - 5 ) !} (81 x)^{21} \cdot 6^{5} + \frac{26 !}{6 ! ( 26 - 6 ) !} (81 x)^{20} \cdot 6^{6} + \frac{26 !}{7 ! ( 26 - 7 ) !} (81 x)^{19} \cdot 6^{7} + \frac{26 !}{8 ! ( 26 - 8 ) !} (81 x)^{18} \cdot 6^{8} + \frac{26 !}{9 ! ( 26 - 9 ) !} (81 x)^{17} \cdot 6^{9} + \frac{26 !}{10 ! ( 26 - 10 ) !} (81 x)^{16} \cdot 6^{10} + \frac{26 !}{11 ! ( 26 - 11 ) !} (81 x)^{15} \cdot 6^{11} + \frac{26 !}{12 ! ( 26 - 12 ) !} (81 x)^{14} \cdot 6^{12} + \frac{26 !}{13 ! ( 26 - 13 ) !} (81 x)^{13} \cdot 6^{13} + \frac{26 !}{14 ! ( 26 - 14 ) !} (81 x)^{12} \cdot 6^{14} + \frac{26 !}{15 ! ( 26 - 15 ) !} (81 x)^{11} \cdot 6^{15} + \frac{26 !}{16 ! ( 26 - 16 ) !} (81 x)^{10} \cdot 6^{16} + \frac{26 !}{17 ! ( 26 - 17 ) !} (81 x)^{9} \cdot 6^{17} + \frac{26 !}{18 ! ( 26 - 18 ) !} (81 x)^{8} \cdot 6^{18} + \frac{26 !}{19 ! ( 26 - 19 ) !} (81 x)^{7} \cdot 6^{19} + \frac{26 !}{20 ! ( 26 - 20 ) !} (81 x)^{6} \cdot 6^{20} + \frac{26 !}{21 ! ( 26 - 21 ) !} (81 x)^{5} \cdot 6^{21} + \frac{26 !}{22 ! ( 26 - 22 ) !} (81 x)^{4} \cdot 6^{22} + \frac{26 !}{23 ! ( 26 - 23 ) !} (81 x)^{3} \cdot 6^{23} + \frac{26 !}{24 ! ( 26 - 24 ) !} (81 x)^{2} \cdot 6^{24} + \frac{26 !}{25 ! ( 26 - 25 ) !} (81 x)^{1} \cdot 6^{25} + \frac{26 !}{26 ! ( 26 - 26 ) !} (81 x)^{0} \cdot 6^{26}

Simplifier

\frac{26 !}{0 ! ( 26 - 0 ) !} (81 x)^{26} \cdot 6^{0} : 8 1^{26} x^{26}

Simplifier

\frac{26 !}{1 ! ( 26 - 1 ) !} (81 x)^{25} \cdot 6^{1} : 8 1^{25} \cdot 156 x^{25}

Simplifier

\frac{26 !}{2 ! ( 26 - 2 ) !} (81 x)^{24} \cdot 6^{2} : 3^{98} \cdot 1300 x^{24}

Simplifier

\frac{26 !}{3 ! ( 26 - 3 ) !} (81 x)^{23} \cdot 6^{3} : 3^{95} \cdot 20800 x^{23}

Simplifier

\frac{26 !}{4 ! ( 26 - 4 ) !} (81 x)^{22} \cdot 6^{4} : 3^{92} \cdot 239200 x^{22}

Simplifier

\frac{26 !}{5 ! ( 26 - 5 ) !} (81 x)^{21} \cdot 6^{5} : 3^{89} \cdot 2104960 x^{21}

Simplifier

\frac{26 !}{6 ! ( 26 - 6 ) !} (81 x)^{20} \cdot 6^{6} : 3^{86} \cdot 14734720 x^{20}

Simplifier

\frac{26 !}{7 ! ( 26 - 7 ) !} (81 x)^{19} \cdot 6^{7} : 3^{83} \cdot 84198400 x^{19}

Simplifier

\frac{26 !}{8 ! ( 26 - 8 ) !} (81 x)^{18} \cdot 6^{8} : 3^{80} \cdot 399942400 x^{18}

Simplifier

\frac{26 !}{9 ! ( 26 - 9 ) !} (81 x)^{17} \cdot 6^{9} : 3^{77} \cdot 1599769600 x^{17}

Simplifier

\frac{26 !}{10 ! ( 26 - 10 ) !} (81 x)^{16} \cdot 6^{10} : 3^{74} \cdot 5439216640 x^{16}

Simplifier

\frac{26 !}{11 ! ( 26 - 11 ) !} (81 x)^{15} \cdot 6^{11} : 3^{71} \cdot 15823175680 x^{15}

Simplifier

\frac{26 !}{12 ! ( 26 - 12 ) !} (81 x)^{14} \cdot 6^{12} : \frac{3 ^{68} \cdot 1.89483 E 19 x ^{14}}{479001600}

Simplifier

\frac{26 !}{13 ! ( 26 - 13 ) !} (81 x)^{13} \cdot 6^{13} : \frac{3 ^{65} \cdot 5.30553 E 20 x ^{13}}{6227020800}

Simplifier

\frac{26 !}{14 ! ( 26 - 14 ) !} (81 x)^{12} \cdot 6^{14} : \frac{3 ^{62} \cdot 7.57933 E 19 x ^{12}}{479001600}

Simplifier

\frac{26 !}{15 ! ( 26 - 15 ) !} (81 x)^{11} \cdot 6^{15} : \frac{3 ^{55} \cdot 1.01058 E 19 x ^{11}}{492800}

Simplifier

\frac{26 !}{16 ! ( 26 - 16 ) !} (81 x)^{10} \cdot 6^{16} : 3^{56} \cdot 348109864960 x^{10}

Simplifier

\frac{26 !}{17 ! ( 26 - 17 ) !} (81 x)^{9} \cdot 6^{17} : 3^{53} \cdot 409541017600 x^{9}

Simplifier

\frac{26 !}{18 ! ( 26 - 18 ) !} (81 x)^{8} \cdot 6^{18} : 3^{50} \cdot 409541017600 x^{8}

Simplifier

\frac{26 !}{19 ! ( 26 - 19 ) !} (81 x)^{7} \cdot 6^{19} : 3^{47} \cdot 344876646400 x^{7}

Simplifier

\frac{26 !}{20 ! ( 26 - 20 ) !} (81 x)^{6} \cdot 6^{20} : 3^{44} \cdot 241413652480 x^{6}

Simplifier

\frac{26 !}{21 ! ( 26 - 21 ) !} (81 x)^{5} \cdot 6^{21} : 3^{41} \cdot 137950658560 x^{5}

Simplifier

\frac{26 !}{22 ! ( 26 - 22 ) !} (81 x)^{4} \cdot 6^{22} : 3^{38} \cdot 62704844800 x^{4}

Simplifier

\frac{26 !}{23 ! ( 26 - 23 ) !} (81 x)^{3} \cdot 6^{23} : 3^{35} \cdot 21810380800 x^{3}

Simplifier

\frac{26 !}{24 ! ( 26 - 24 ) !} (81 x)^{2} \cdot 6^{24} : 3^{32} \cdot 5452595200 x^{2}

Simplifier

\frac{26 !}{25 ! ( 26 - 25 ) !} (81 x)^{1} \cdot 6^{25} : 6^{25} \cdot 2106 x

Simplifier

\frac{26 !}{26 ! ( 26 - 26 ) !} (81 x)^{0} \cdot 6^{26} : 6^{26}

= 8 1^{26} x^{26} + 8 1^{25} \cdot 156 x^{25} + 3^{98} \cdot 1300 x^{24} + 3^{95} \cdot 20800 x^{23} + 3^{92} \cdot 239200 x^{22} + 3^{89} \cdot 2104960 x^{21} + 3^{86} \cdot 14734720 x^{20} + 3^{83} \cdot 84198400 x^{19} + 3^{80} \cdot 399942400 x^{18} + 3^{77} \cdot 1599769600 x^{17} + 3^{74} \cdot 5439216640 x^{16} + 3^{71} \cdot 15823175680 x^{15} + \frac{3 ^{68} \cdot 1.89483 E 19 x ^{14}}{479001600} + \frac{3 ^{65} \cdot 5.30553 E 20 x ^{13}}{6227020800} + \frac{3 ^{62} \cdot 7.57933 E 19 x ^{12}}{479001600} + \frac{3 ^{55} \cdot 1.01058 E 19 x ^{11}}{492800} + 3^{56} \cdot 348109864960 x^{10} + 3^{53} \cdot 409541017600 x^{9} + 3^{50} \cdot 409541017600 x^{8} + 3^{47} \cdot 344876646400 x^{7} + 3^{44} \cdot 241413652480 x^{6} + 3^{41} \cdot 137950658560 x^{5} + 3^{38} \cdot 62704844800 x^{4} + 3^{35} \cdot 21810380800 x^{3} + 3^{32} \cdot 5452595200 x^{2} + 6^{25} \cdot 2106 x + 6^{26}

\frac{3 ^{68} \cdot 1.89483 E 19 x ^{14}}{479001600} = 3^{68} \cdot 39557939200 x^{14}

\frac{3 ^{65} \cdot 5.30553 E 20 x ^{13}}{6227020800} = \frac{3 ^{60} \cdot 7.57933 E 19 x ^{13}}{3660800}

\frac{3 ^{62} \cdot 7.57933 E 19 x ^{12}}{479001600} = \frac{3 ^{57} \cdot 1.08276 E 19 x ^{12}}{281600}

\frac{3 ^{55} \cdot 1.01058 E 19 x ^{11}}{492800} = 3^{55} \cdot 2.05068 E 13 x^{11}

= 8 1^{26} x^{26} + 8 1^{25} \cdot 156 x^{25} + 3^{98} \cdot 1300 x^{24} + 3^{95} \cdot 20800 x^{23} + 3^{92} \cdot 239200 x^{22} + 3^{89} \cdot 2104960 x^{21} + 3^{86} \cdot 14734720 x^{20} + 3^{83} \cdot 84198400 x^{19} + 3^{80} \cdot 399942400 x^{18} + 3^{77} \cdot 1599769600 x^{17} + 3^{74} \cdot 5439216640 x^{16} + 3^{71} \cdot 15823175680 x^{15} + 3^{68} \cdot 39557939200 x^{14} + \frac{3 ^{60} \cdot 7.57933 E 19 x ^{13}}{3660800} + \frac{3 ^{57} \cdot 1.08276 E 19 x ^{12}}{281600} + 3^{55} \cdot 2.05068 E 13 x^{11} + 3^{56} \cdot 348109864960 x^{10} + 3^{53} \cdot 409541017600 x^{9} + 3^{50} \cdot 409541017600 x^{8} + 3^{47} \cdot 344876646400 x^{7} + 3^{44} \cdot 241413652480 x^{6} + 3^{41} \cdot 137950658560 x^{5} + 3^{38} \cdot 62704844800 x^{4} + 3^{35} \cdot 21810380800 x^{3} + 3^{32} \cdot 5452595200 x^{2} + 6^{25} \cdot 2106 x + 6^{26}

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