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Phổ biến Đại số >

khai triển (7-\sqrt[5]{x/(282475249)})^{17}

Lời Giải

khai triển

(7 - 5 \frac{x}{282475249})^{17}

Lời Giải

7^{17} - 7^{14} \cdot 17 5 x + 7^{15} \cdot 136 (5 \frac{x}{282475249})^{2} - 7^{14} \cdot 680 (5 \frac{x}{282475249})^{3} + 7^{13} \cdot 2380 (5 \frac{x}{282475249})^{4} - 303212 x + 86632 x 5 \frac{x}{282475249} - \frac{19448 x 5 x ^{2}}{5 28247524 9 ^{2}} + \frac{24310 x 5 x ^{3}}{7 5 28247524 9 ^{3}} - \frac{24310 x 5 x ^{4}}{49 5 28247524 9 ^{4}} + \frac{19448 x ^{2}}{96889010407} - \frac{1768 x ^{2} 5 x}{823543 \cdot 34 3 ^{2} 5 282475249} + \frac{884 x ^{2} 5 x ^{2}}{117649 \cdot 240 1 ^{2} 5 28247524 9 ^{2}} - \frac{340 x ^{2} 5 x ^{3}}{1680 7 ^{3} 5 28247524 9 ^{3}} + \frac{680 x ^{2} 5 x ^{4}}{34 3 ^{5} \cdot 49 5 28247524 9 ^{4}} - \frac{136 x ^{3}}{4 9 ^{14}} + \frac{17 x ^{3} 5 x}{7 ^{29} 5 282475249} - \frac{x ^{3} ( 5 \frac{x}{282475249} ) ^{2}}{28247524 9 ^{3}}

Các bước giải pháp

(7 - 5 \frac{x}{282475249})^{17}

Áp dụng định lý nhị thức:

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

a = 7, b = - 5 \frac{x}{282475249}

= i = 0 \sum 17 (i 17) \cdot 7^{(17 - i)} (- 5 \frac{x}{282475249})^{i}

Khai triển phép tổng

= \frac{17 !}{0 ! ( 17 - 0 ) !} \cdot 7^{17} (- 5 \frac{x}{282475249})^{0} + \frac{17 !}{1 ! ( 17 - 1 ) !} \cdot 7^{16} (- 5 \frac{x}{282475249})^{1} + \frac{17 !}{2 ! ( 17 - 2 ) !} \cdot 7^{15} (- 5 \frac{x}{282475249})^{2} + \frac{17 !}{3 ! ( 17 - 3 ) !} \cdot 7^{14} (- 5 \frac{x}{282475249})^{3} + \frac{17 !}{4 ! ( 17 - 4 ) !} \cdot 7^{13} (- 5 \frac{x}{282475249})^{4} + \frac{17 !}{5 ! ( 17 - 5 ) !} \cdot 7^{12} (- 5 \frac{x}{282475249})^{5} + \frac{17 !}{6 ! ( 17 - 6 ) !} \cdot 7^{11} (- 5 \frac{x}{282475249})^{6} + \frac{17 !}{7 ! ( 17 - 7 ) !} \cdot 7^{10} (- 5 \frac{x}{282475249})^{7} + \frac{17 !}{8 ! ( 17 - 8 ) !} \cdot 7^{9} (- 5 \frac{x}{282475249})^{8} + \frac{17 !}{9 ! ( 17 - 9 ) !} \cdot 7^{8} (- 5 \frac{x}{282475249})^{9} + \frac{17 !}{10 ! ( 17 - 10 ) !} \cdot 7^{7} (- 5 \frac{x}{282475249})^{10} + \frac{17 !}{11 ! ( 17 - 11 ) !} \cdot 7^{6} (- 5 \frac{x}{282475249})^{11} + \frac{17 !}{12 ! ( 17 - 12 ) !} \cdot 7^{5} (- 5 \frac{x}{282475249})^{12} + \frac{17 !}{13 ! ( 17 - 13 ) !} \cdot 7^{4} (- 5 \frac{x}{282475249})^{13} + \frac{17 !}{14 ! ( 17 - 14 ) !} \cdot 7^{3} (- 5 \frac{x}{282475249})^{14} + \frac{17 !}{15 ! ( 17 - 15 ) !} \cdot 7^{2} (- 5 \frac{x}{282475249})^{15} + \frac{17 !}{16 ! ( 17 - 16 ) !} \cdot 7^{1} (- 5 \frac{x}{282475249})^{16} + \frac{17 !}{17 ! ( 17 - 17 ) !} \cdot 7^{0} (- 5 \frac{x}{282475249})^{17}

Rút gọn

\frac{17 !}{0 ! ( 17 - 0 ) !} \cdot 7^{17} (- 5 \frac{x}{282475249})^{0} : 7^{17}

Rút gọn

\frac{17 !}{1 ! ( 17 - 1 ) !} \cdot 7^{16} (- 5 \frac{x}{282475249})^{1} : - \frac{7 ^{16} \cdot 17 5 x}{5 282475249}

Rút gọn

\frac{17 !}{2 ! ( 17 - 2 ) !} \cdot 7^{15} (- 5 \frac{x}{282475249})^{2} : 7^{15} \cdot 136 (5 \frac{x}{282475249})^{2}

Rút gọn

\frac{17 !}{3 ! ( 17 - 3 ) !} \cdot 7^{14} (- 5 \frac{x}{282475249})^{3} : - 7^{14} \cdot 680 (5 \frac{x}{282475249})^{3}

Rút gọn

\frac{17 !}{4 ! ( 17 - 4 ) !} \cdot 7^{13} (- 5 \frac{x}{282475249})^{4} : 7^{13} \cdot 2380 (5 \frac{x}{282475249})^{4}

Rút gọn

\frac{17 !}{5 ! ( 17 - 5 ) !} \cdot 7^{12} (- 5 \frac{x}{282475249})^{5} : - 303212 x

Rút gọn

\frac{17 !}{6 ! ( 17 - 6 ) !} \cdot 7^{11} (- 5 \frac{x}{282475249})^{6} : 7^{11} \cdot 12376 (5 \frac{x}{282475249})^{6}

Rút gọn

\frac{17 !}{7 ! ( 17 - 7 ) !} \cdot 7^{10} (- 5 \frac{x}{282475249})^{7} : - 5493578642552 (5 \frac{x}{282475249})^{7}

Rút gọn

\frac{17 !}{8 ! ( 17 - 8 ) !} \cdot 7^{9} (- 5 \frac{x}{282475249})^{8} : 980996186170 (5 \frac{x}{282475249})^{8}

Rút gọn

\frac{17 !}{9 ! ( 17 - 9 ) !} \cdot 7^{8} (- 5 \frac{x}{282475249})^{9} : - 140142312310 (5 \frac{x}{282475249})^{9}

Rút gọn

\frac{17 !}{10 ! ( 17 - 10 ) !} \cdot 7^{7} (- 5 \frac{x}{282475249})^{10} : \frac{16016264264 x ^{2}}{28247524 9 ^{2}}

Rút gọn

\frac{17 !}{11 ! ( 17 - 11 ) !} \cdot 7^{6} (- 5 \frac{x}{282475249})^{11} : - 1456024024 (5 \frac{x}{282475249})^{11}

Rút gọn

\frac{17 !}{12 ! ( 17 - 12 ) !} \cdot 7^{5} (- 5 \frac{x}{282475249})^{12} : 104001716 (5 \frac{x}{282475249})^{12}

Rút gọn

\frac{17 !}{13 ! ( 17 - 13 ) !} \cdot 7^{4} (- 5 \frac{x}{282475249})^{13} : - 5714380 (5 \frac{x}{282475249})^{13}

Rút gọn

\frac{17 !}{14 ! ( 17 - 14 ) !} \cdot 7^{3} (- 5 \frac{x}{282475249})^{14} : 233240 (5 \frac{x}{282475249})^{14}

Rút gọn

\frac{17 !}{15 ! ( 17 - 15 ) !} \cdot 7^{2} (- 5 \frac{x}{282475249})^{15} : - \frac{6664 x ^{3}}{28247524 9 ^{3}}

Rút gọn

\frac{17 !}{16 ! ( 17 - 16 ) !} \cdot 7^{1} (- 5 \frac{x}{282475249})^{16} : 119 (5 \frac{x}{282475249})^{16}

Rút gọn

\frac{17 !}{17 ! ( 17 - 17 ) !} \cdot 7^{0} (- 5 \frac{x}{282475249})^{17} : - (5 \frac{x}{282475249})^{17}

= 7^{17} - \frac{7 ^{16} \cdot 17 5 x}{5 282475249} + 7^{15} \cdot 136 (5 \frac{x}{282475249})^{2} - 7^{14} \cdot 680 (5 \frac{x}{282475249})^{3} + 7^{13} \cdot 2380 (5 \frac{x}{282475249})^{4} - 303212 x + 7^{11} \cdot 12376 (5 \frac{x}{282475249})^{6} - 5493578642552 (5 \frac{x}{282475249})^{7} + 980996186170 (5 \frac{x}{282475249})^{8} - 140142312310 (5 \frac{x}{282475249})^{9} + \frac{16016264264 x ^{2}}{28247524 9 ^{2}} - 1456024024 (5 \frac{x}{282475249})^{11} + 104001716 (5 \frac{x}{282475249})^{12} - 5714380 (5 \frac{x}{282475249})^{13} + 233240 (5 \frac{x}{282475249})^{14} - \frac{6664 x ^{3}}{28247524 9 ^{3}} + 119 (5 \frac{x}{282475249})^{16} - (5 \frac{x}{282475249})^{17}

Rút gọn

7^{17} - \frac{7 ^{16} \cdot 17 5 x}{5 282475249} + 7^{15} \cdot 136 (5 \frac{x}{282475249})^{2} - 7^{14} \cdot 680 (5 \frac{x}{282475249})^{3} + 7^{13} \cdot 2380 (5 \frac{x}{282475249})^{4} - 303212 x + 7^{11} \cdot 12376 (5 \frac{x}{282475249})^{6} - 5493578642552 (5 \frac{x}{282475249})^{7} + 980996186170 (5 \frac{x}{282475249})^{8} - 140142312310 (5 \frac{x}{282475249})^{9} + \frac{16016264264 x ^{2}}{28247524 9 ^{2}} - 1456024024 (5 \frac{x}{282475249})^{11} + 104001716 (5 \frac{x}{282475249})^{12} - 5714380 (5 \frac{x}{282475249})^{13} + 233240 (5 \frac{x}{282475249})^{14} - \frac{6664 x ^{3}}{28247524 9 ^{3}} + 119 (5 \frac{x}{282475249})^{16} - (5 \frac{x}{282475249})^{17} : 7^{17} - 7^{14} \cdot 17 5 x + 7^{15} \cdot 136 (5 \frac{x}{282475249})^{2} - 7^{14} \cdot 680 (5 \frac{x}{282475249})^{3} + 7^{13} \cdot 2380 (5 \frac{x}{282475249})^{4} - 303212 x + 86632 x 5 \frac{x}{282475249} - \frac{19448 x 5 x ^{2}}{5 28247524 9 ^{2}} + \frac{24310 x 5 x ^{3}}{7 5 28247524 9 ^{3}} - \frac{24310 x 5 x ^{4}}{49 5 28247524 9 ^{4}} + \frac{19448 x ^{2}}{96889010407} - \frac{1768 x ^{2} 5 x}{823543 \cdot 34 3 ^{2} 5 282475249} + \frac{884 x ^{2} 5 x ^{2}}{117649 \cdot 240 1 ^{2} 5 28247524 9 ^{2}} - \frac{340 x ^{2} 5 x ^{3}}{1680 7 ^{3} 5 28247524 9 ^{3}} + \frac{680 x ^{2} 5 x ^{4}}{34 3 ^{5} \cdot 49 5 28247524 9 ^{4}} - \frac{136 x ^{3}}{4 9 ^{14}} + \frac{17 x ^{3} 5 x}{7 ^{29} 5 282475249} - \frac{x ^{3} ( 5 \frac{x}{282475249} ) ^{2}}{28247524 9 ^{3}}

= 7^{17} - 7^{14} \cdot 17 5 x + 7^{15} \cdot 136 (5 \frac{x}{282475249})^{2} - 7^{14} \cdot 680 (5 \frac{x}{282475249})^{3} + 7^{13} \cdot 2380 (5 \frac{x}{282475249})^{4} - 303212 x + 86632 x 5 \frac{x}{282475249} - \frac{19448 x 5 x ^{2}}{5 28247524 9 ^{2}} + \frac{24310 x 5 x ^{3}}{7 5 28247524 9 ^{3}} - \frac{24310 x 5 x ^{4}}{49 5 28247524 9 ^{4}} + \frac{19448 x ^{2}}{96889010407} - \frac{1768 x ^{2} 5 x}{823543 \cdot 34 3 ^{2} 5 282475249} + \frac{884 x ^{2} 5 x ^{2}}{117649 \cdot 240 1 ^{2} 5 28247524 9 ^{2}} - \frac{340 x ^{2} 5 x ^{3}}{1680 7 ^{3} 5 28247524 9 ^{3}} + \frac{680 x ^{2} 5 x ^{4}}{34 3 ^{5} \cdot 49 5 28247524 9 ^{4}} - \frac{136 x ^{3}}{4 9 ^{14}} + \frac{17 x ^{3} 5 x}{7 ^{29} 5 282475249} - \frac{x ^{3} ( 5 \frac{x}{282475249} ) ^{2}}{28247524 9 ^{3}}

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