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Populaire Algèbre >

développer (7-\sqrt[3]{x/(13841287201)})^{17}

Solution

développer

(7 - 3 \frac{x}{13841287201})^{17}

Solution

7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - 33320 x + 16660 x 3 \frac{x}{13841287201} - \frac{6188 x 3 x ^{2}}{3 1384128720 1 ^{2}} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - \frac{19448 3 x ^{7}}{282475249 3 1384128720 1 ^{2}} + \frac{980996186170 x ^{2} 3 x ^{2}}{1384128720 1 ^{2} 3 1384128720 1 ^{2}} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + \frac{6670664 3 x ^{10}}{1384128720 1 ^{3}} - \frac{1456024024 x ^{3} 3 x ^{2}}{1384128720 1 ^{3} 3 1384128720 1 ^{2}} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - \frac{2380 3 x ^{13}}{1384128720 1 ^{4}} + \frac{233240 x ^{4} 3 x ^{2}}{1384128720 1 ^{4} 3 1384128720 1 ^{2}} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + \frac{17 3 x ^{16}}{1384128720 1 ^{5} \cdot 343} - \frac{x ^{5} ( 3 \frac{x}{13841287201} ) ^{2}}{1384128720 1 ^{5}}

étapes des solutions

(7 - 3 \frac{x}{13841287201})^{17}

Appliquer le théorème du binôme:

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

a = 7, b = - 3 \frac{x}{13841287201}

= i = 0 \sum 17 (i 17) \cdot 7^{(17 - i)} (- 3 \frac{x}{13841287201})^{i}

Développer la somme

= \frac{17 !}{0 ! ( 17 - 0 ) !} \cdot 7^{17} (- 3 \frac{x}{13841287201})^{0} + \frac{17 !}{1 ! ( 17 - 1 ) !} \cdot 7^{16} (- 3 \frac{x}{13841287201})^{1} + \frac{17 !}{2 ! ( 17 - 2 ) !} \cdot 7^{15} (- 3 \frac{x}{13841287201})^{2} + \frac{17 !}{3 ! ( 17 - 3 ) !} \cdot 7^{14} (- 3 \frac{x}{13841287201})^{3} + \frac{17 !}{4 ! ( 17 - 4 ) !} \cdot 7^{13} (- 3 \frac{x}{13841287201})^{4} + \frac{17 !}{5 ! ( 17 - 5 ) !} \cdot 7^{12} (- 3 \frac{x}{13841287201})^{5} + \frac{17 !}{6 ! ( 17 - 6 ) !} \cdot 7^{11} (- 3 \frac{x}{13841287201})^{6} + \frac{17 !}{7 ! ( 17 - 7 ) !} \cdot 7^{10} (- 3 \frac{x}{13841287201})^{7} + \frac{17 !}{8 ! ( 17 - 8 ) !} \cdot 7^{9} (- 3 \frac{x}{13841287201})^{8} + \frac{17 !}{9 ! ( 17 - 9 ) !} \cdot 7^{8} (- 3 \frac{x}{13841287201})^{9} + \frac{17 !}{10 ! ( 17 - 10 ) !} \cdot 7^{7} (- 3 \frac{x}{13841287201})^{10} + \frac{17 !}{11 ! ( 17 - 11 ) !} \cdot 7^{6} (- 3 \frac{x}{13841287201})^{11} + \frac{17 !}{12 ! ( 17 - 12 ) !} \cdot 7^{5} (- 3 \frac{x}{13841287201})^{12} + \frac{17 !}{13 ! ( 17 - 13 ) !} \cdot 7^{4} (- 3 \frac{x}{13841287201})^{13} + \frac{17 !}{14 ! ( 17 - 14 ) !} \cdot 7^{3} (- 3 \frac{x}{13841287201})^{14} + \frac{17 !}{15 ! ( 17 - 15 ) !} \cdot 7^{2} (- 3 \frac{x}{13841287201})^{15} + \frac{17 !}{16 ! ( 17 - 16 ) !} \cdot 7^{1} (- 3 \frac{x}{13841287201})^{16} + \frac{17 !}{17 ! ( 17 - 17 ) !} \cdot 7^{0} (- 3 \frac{x}{13841287201})^{17}

Simplifier

\frac{17 !}{0 ! ( 17 - 0 ) !} \cdot 7^{17} (- 3 \frac{x}{13841287201})^{0} : 7^{17}

Simplifier

\frac{17 !}{1 ! ( 17 - 1 ) !} \cdot 7^{16} (- 3 \frac{x}{13841287201})^{1} : - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201}

Simplifier

\frac{17 !}{2 ! ( 17 - 2 ) !} \cdot 7^{15} (- 3 \frac{x}{13841287201})^{2} : 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2}

Simplifier

\frac{17 !}{3 ! ( 17 - 3 ) !} \cdot 7^{14} (- 3 \frac{x}{13841287201})^{3} : - \frac{7 ^{14} \cdot 680 x}{13841287201}

Simplifier

\frac{17 !}{4 ! ( 17 - 4 ) !} \cdot 7^{13} (- 3 \frac{x}{13841287201})^{4} : 7^{13} \cdot 2380 (3 \frac{x}{13841287201})^{4}

Simplifier

\frac{17 !}{5 ! ( 17 - 5 ) !} \cdot 7^{12} (- 3 \frac{x}{13841287201})^{5} : - 7^{12} \cdot 6188 (3 \frac{x}{13841287201})^{5}

Simplifier

\frac{17 !}{6 ! ( 17 - 6 ) !} \cdot 7^{11} (- 3 \frac{x}{13841287201})^{6} : \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}}

Simplifier

\frac{17 !}{7 ! ( 17 - 7 ) !} \cdot 7^{10} (- 3 \frac{x}{13841287201})^{7} : - 5493578642552 (3 \frac{x}{13841287201})^{7}

Simplifier

\frac{17 !}{8 ! ( 17 - 8 ) !} \cdot 7^{9} (- 3 \frac{x}{13841287201})^{8} : 980996186170 (3 \frac{x}{13841287201})^{8}

Simplifier

\frac{17 !}{9 ! ( 17 - 9 ) !} \cdot 7^{8} (- 3 \frac{x}{13841287201})^{9} : - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}}

Simplifier

\frac{17 !}{10 ! ( 17 - 10 ) !} \cdot 7^{7} (- 3 \frac{x}{13841287201})^{10} : 16016264264 (3 \frac{x}{13841287201})^{10}

Simplifier

\frac{17 !}{11 ! ( 17 - 11 ) !} \cdot 7^{6} (- 3 \frac{x}{13841287201})^{11} : - 1456024024 (3 \frac{x}{13841287201})^{11}

Simplifier

\frac{17 !}{12 ! ( 17 - 12 ) !} \cdot 7^{5} (- 3 \frac{x}{13841287201})^{12} : \frac{104001716 x ^{4}}{1384128720 1 ^{4}}

Simplifier

\frac{17 !}{13 ! ( 17 - 13 ) !} \cdot 7^{4} (- 3 \frac{x}{13841287201})^{13} : - 5714380 (3 \frac{x}{13841287201})^{13}

Simplifier

\frac{17 !}{14 ! ( 17 - 14 ) !} \cdot 7^{3} (- 3 \frac{x}{13841287201})^{14} : 233240 (3 \frac{x}{13841287201})^{14}

Simplifier

\frac{17 !}{15 ! ( 17 - 15 ) !} \cdot 7^{2} (- 3 \frac{x}{13841287201})^{15} : - \frac{6664 x ^{5}}{1384128720 1 ^{5}}

Simplifier

\frac{17 !}{16 ! ( 17 - 16 ) !} \cdot 7^{1} (- 3 \frac{x}{13841287201})^{16} : 119 (3 \frac{x}{13841287201})^{16}

Simplifier

\frac{17 !}{17 ! ( 17 - 17 ) !} \cdot 7^{0} (- 3 \frac{x}{13841287201})^{17} : - (3 \frac{x}{13841287201})^{17}

= 7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - \frac{7 ^{14} \cdot 680 x}{13841287201} + 7^{13} \cdot 2380 (3 \frac{x}{13841287201})^{4} - 7^{12} \cdot 6188 (3 \frac{x}{13841287201})^{5} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - 5493578642552 (3 \frac{x}{13841287201})^{7} + 980996186170 (3 \frac{x}{13841287201})^{8} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + 16016264264 (3 \frac{x}{13841287201})^{10} - 1456024024 (3 \frac{x}{13841287201})^{11} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - 5714380 (3 \frac{x}{13841287201})^{13} + 233240 (3 \frac{x}{13841287201})^{14} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + 119 (3 \frac{x}{13841287201})^{16} - (3 \frac{x}{13841287201})^{17}

Simplifier

7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - \frac{7 ^{14} \cdot 680 x}{13841287201} + 7^{13} \cdot 2380 (3 \frac{x}{13841287201})^{4} - 7^{12} \cdot 6188 (3 \frac{x}{13841287201})^{5} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - 5493578642552 (3 \frac{x}{13841287201})^{7} + 980996186170 (3 \frac{x}{13841287201})^{8} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + 16016264264 (3 \frac{x}{13841287201})^{10} - 1456024024 (3 \frac{x}{13841287201})^{11} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - 5714380 (3 \frac{x}{13841287201})^{13} + 233240 (3 \frac{x}{13841287201})^{14} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + 119 (3 \frac{x}{13841287201})^{16} - (3 \frac{x}{13841287201})^{17} : 7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - 33320 x + 16660 x 3 \frac{x}{13841287201} - \frac{6188 x 3 x ^{2}}{3 1384128720 1 ^{2}} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - \frac{19448 x ^{\frac{7}{3}}}{282475249 \cdot 1384128720 1 ^{\frac{2}{3}}} + \frac{980996186170 x ^{2} 3 x ^{2}}{1384128720 1 ^{2} 3 1384128720 1 ^{2}} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + \frac{6670664 x ^{\frac{10}{3}}}{1384128720 1 ^{3}} - \frac{1456024024 x ^{3} 3 x ^{2}}{1384128720 1 ^{3} 3 1384128720 1 ^{2}} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - \frac{2380 x ^{\frac{13}{3}}}{1384128720 1 ^{4}} + \frac{233240 x ^{4} 3 x ^{2}}{1384128720 1 ^{4} 3 1384128720 1 ^{2}} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + \frac{17 x ^{\frac{16}{3}}}{1384128720 1 ^{5} \cdot 343} - \frac{x ^{5} ( 3 \frac{x}{13841287201} ) ^{2}}{1384128720 1 ^{5}}

= 7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - 33320 x + 16660 x 3 \frac{x}{13841287201} - \frac{6188 x 3 x ^{2}}{3 1384128720 1 ^{2}} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - \frac{19448 x ^{\frac{7}{3}}}{282475249 \cdot 1384128720 1 ^{\frac{2}{3}}} + \frac{980996186170 x ^{2} 3 x ^{2}}{1384128720 1 ^{2} 3 1384128720 1 ^{2}} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + \frac{6670664 x ^{\frac{10}{3}}}{1384128720 1 ^{3}} - \frac{1456024024 x ^{3} 3 x ^{2}}{1384128720 1 ^{3} 3 1384128720 1 ^{2}} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - \frac{2380 x ^{\frac{13}{3}}}{1384128720 1 ^{4}} + \frac{233240 x ^{4} 3 x ^{2}}{1384128720 1 ^{4} 3 1384128720 1 ^{2}} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + \frac{17 x ^{\frac{16}{3}}}{1384128720 1 ^{5} \cdot 343} - \frac{x ^{5} ( 3 \frac{x}{13841287201} ) ^{2}}{1384128720 1 ^{5}}

Simplifier

7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - 33320 x + 16660 x 3 \frac{x}{13841287201} - \frac{6188 x 3 x ^{2}}{3 1384128720 1 ^{2}} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - \frac{19448 x ^{\frac{7}{3}}}{282475249 \cdot 1384128720 1 ^{\frac{2}{3}}} + \frac{980996186170 x ^{2} 3 x ^{2}}{1384128720 1 ^{2} 3 1384128720 1 ^{2}} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + \frac{6670664 x ^{\frac{10}{3}}}{1384128720 1 ^{3}} - \frac{1456024024 x ^{3} 3 x ^{2}}{1384128720 1 ^{3} 3 1384128720 1 ^{2}} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - \frac{2380 x ^{\frac{13}{3}}}{1384128720 1 ^{4}} + \frac{233240 x ^{4} 3 x ^{2}}{1384128720 1 ^{4} 3 1384128720 1 ^{2}} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + \frac{17 x ^{\frac{16}{3}}}{1384128720 1 ^{5} \cdot 343} - \frac{x ^{5} ( 3 \frac{x}{13841287201} ) ^{2}}{1384128720 1 ^{5}} : 7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - 33320 x + 16660 x 3 \frac{x}{13841287201} - \frac{6188 x 3 x ^{2}}{3 1384128720 1 ^{2}} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - \frac{19448 3 x ^{7}}{282475249 3 1384128720 1 ^{2}} + \frac{980996186170 x ^{2} 3 x ^{2}}{1384128720 1 ^{2} 3 1384128720 1 ^{2}} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + \frac{6670664 3 x ^{10}}{1384128720 1 ^{3}} - \frac{1456024024 x ^{3} 3 x ^{2}}{1384128720 1 ^{3} 3 1384128720 1 ^{2}} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - \frac{2380 3 x ^{13}}{1384128720 1 ^{4}} + \frac{233240 x ^{4} 3 x ^{2}}{1384128720 1 ^{4} 3 1384128720 1 ^{2}} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + \frac{17 3 x ^{16}}{1384128720 1 ^{5} \cdot 343} - \frac{x ^{5} ( 3 \frac{x}{13841287201} ) ^{2}}{1384128720 1 ^{5}}

= 7^{17} - \frac{7 ^{16} \cdot 17 3 x}{3 13841287201} + 7^{15} \cdot 136 (3 \frac{x}{13841287201})^{2} - 33320 x + 16660 x 3 \frac{x}{13841287201} - \frac{6188 x 3 x ^{2}}{3 1384128720 1 ^{2}} + \frac{7 ^{11} \cdot 12376 x ^{2}}{1384128720 1 ^{2}} - \frac{19448 3 x ^{7}}{282475249 3 1384128720 1 ^{2}} + \frac{980996186170 x ^{2} 3 x ^{2}}{1384128720 1 ^{2} 3 1384128720 1 ^{2}} - \frac{140142312310 x ^{3}}{1384128720 1 ^{3}} + \frac{6670664 3 x ^{10}}{1384128720 1 ^{3}} - \frac{1456024024 x ^{3} 3 x ^{2}}{1384128720 1 ^{3} 3 1384128720 1 ^{2}} + \frac{104001716 x ^{4}}{1384128720 1 ^{4}} - \frac{2380 3 x ^{13}}{1384128720 1 ^{4}} + \frac{233240 x ^{4} 3 x ^{2}}{1384128720 1 ^{4} 3 1384128720 1 ^{2}} - \frac{6664 x ^{5}}{1384128720 1 ^{5}} + \frac{17 3 x ^{16}}{1384128720 1 ^{5} \cdot 343} - \frac{x ^{5} ( 3 \frac{x}{13841287201} ) ^{2}}{1384128720 1 ^{5}}

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