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인기 있는 대수학 >

확장 (7-\sqrt[5]{x/(96889010407)})^{20}

해법

확장

(7 - 5 \frac{x}{96889010407})^{20}

해법

7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - 759696 x + 271320 x 5 \frac{x}{96889010407} - \frac{77520 x 5 x ^{2}}{5 9688901040 7 ^{2}} + \frac{125970 x 5 x ^{3}}{7 5 9688901040 7 ^{3}} - \frac{167960 x 5 x ^{4}}{49 5 9688901040 7 ^{4}} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - \frac{403271960 5 240 1 ^{3} 5 x ^{11}}{9688901040 7 ^{2}} + \frac{726191981970 x ^{2} 5 x ^{2}}{9688901040 7 ^{2} 5 9688901040 7 ^{2}} - \frac{63841053360 x ^{2} 5 x ^{3}}{9688901040 7 ^{2} 5 9688901040 7 ^{3}} + \frac{4560075240 x ^{2} 5 x ^{4}}{9688901040 7 ^{2} 5 9688901040 7 ^{4}} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + \frac{33915 5 49 5 x ^{16}}{9688901040 7 ^{3}} - \frac{391020 x ^{3} 5 x ^{2}}{9688901040 7 ^{3} 5 9688901040 7 ^{2}} + \frac{9310 x ^{3} 5 x ^{3}}{9688901040 7 ^{3} 5 9688901040 7 ^{3}} - \frac{140 x ^{3} 5 x ^{4}}{9688901040 7 ^{3} 5 9688901040 7 ^{4}} + \frac{x ^{4}}{9688901040 7 ^{4}}

솔루션 단계

(7 - 5 \frac{x}{96889010407})^{20}

이항 정리 적용:

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

a = 7, b = - 5 \frac{x}{96889010407}

= i = 0 \sum 20 (i 20) \cdot 7^{(20 - i)} (- 5 \frac{x}{96889010407})^{i}

합계 확장

= \frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 7^{20} (- 5 \frac{x}{96889010407})^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 7^{19} (- 5 \frac{x}{96889010407})^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 7^{18} (- 5 \frac{x}{96889010407})^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 7^{17} (- 5 \frac{x}{96889010407})^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 7^{16} (- 5 \frac{x}{96889010407})^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 7^{15} (- 5 \frac{x}{96889010407})^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 7^{14} (- 5 \frac{x}{96889010407})^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 7^{13} (- 5 \frac{x}{96889010407})^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 7^{12} (- 5 \frac{x}{96889010407})^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 7^{11} (- 5 \frac{x}{96889010407})^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 7^{10} (- 5 \frac{x}{96889010407})^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 7^{9} (- 5 \frac{x}{96889010407})^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 7^{8} (- 5 \frac{x}{96889010407})^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 7^{7} (- 5 \frac{x}{96889010407})^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 7^{6} (- 5 \frac{x}{96889010407})^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 7^{5} (- 5 \frac{x}{96889010407})^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 7^{4} (- 5 \frac{x}{96889010407})^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 7^{3} (- 5 \frac{x}{96889010407})^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 7^{2} (- 5 \frac{x}{96889010407})^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 7^{1} (- 5 \frac{x}{96889010407})^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 7^{0} (- 5 \frac{x}{96889010407})^{20}

\frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 7^{20} (- 5 \frac{x}{96889010407})^{0}

간소화하다 :

7^{20}

\frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 7^{19} (- 5 \frac{x}{96889010407})^{1}

간소화하다 :

- \frac{7 ^{19} \cdot 20 5 x}{5 96889010407}

\frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 7^{18} (- 5 \frac{x}{96889010407})^{2}

간소화하다 :

7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2}

\frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 7^{17} (- 5 \frac{x}{96889010407})^{3}

간소화하다 :

- 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3}

\frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 7^{16} (- 5 \frac{x}{96889010407})^{4}

간소화하다 :

7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4}

\frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 7^{15} (- 5 \frac{x}{96889010407})^{5}

간소화하다 :

- \frac{7 ^{15} \cdot 15504 x}{96889010407}

\frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 7^{14} (- 5 \frac{x}{96889010407})^{6}

간소화하다 :

7^{14} \cdot 38760 (5 \frac{x}{96889010407})^{6}

\frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 7^{13} (- 5 \frac{x}{96889010407})^{7}

간소화하다 :

- 7^{13} \cdot 77520 (5 \frac{x}{96889010407})^{7}

\frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 7^{12} (- 5 \frac{x}{96889010407})^{8}

간소화하다 :

7^{12} \cdot 125970 (5 \frac{x}{96889010407})^{8}

\frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 7^{11} (- 5 \frac{x}{96889010407})^{9}

간소화하다 :

- 7^{11} \cdot 167960 (5 \frac{x}{96889010407})^{9}

\frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 7^{10} (- 5 \frac{x}{96889010407})^{10}

간소화하다 :

\frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800}

\frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 7^{9} (- 5 \frac{x}{96889010407})^{11}

간소화하다 :

- 6777791831720 (5 \frac{x}{96889010407})^{11}

\frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 7^{8} (- 5 \frac{x}{96889010407})^{12}

간소화하다 :

726191981970 (5 \frac{x}{96889010407})^{12}

\frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 7^{7} (- 5 \frac{x}{96889010407})^{13}

간소화하다 :

- 63841053360 (5 \frac{x}{96889010407})^{13}

\frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 7^{6} (- 5 \frac{x}{96889010407})^{14}

간소화하다 :

4560075240 (5 \frac{x}{96889010407})^{14}

\frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 7^{5} (- 5 \frac{x}{96889010407})^{15}

간소화하다 :

- \frac{260575728 x ^{3}}{9688901040 7 ^{3}}

\frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 7^{4} (- 5 \frac{x}{96889010407})^{16}

간소화하다 :

11632845 (5 \frac{x}{96889010407})^{16}

\frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 7^{3} (- 5 \frac{x}{96889010407})^{17}

간소화하다 :

- 391020 (5 \frac{x}{96889010407})^{17}

\frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 7^{2} (- 5 \frac{x}{96889010407})^{18}

간소화하다 :

9310 (5 \frac{x}{96889010407})^{18}

\frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 7^{1} (- 5 \frac{x}{96889010407})^{19}

간소화하다 :

- 140 (5 \frac{x}{96889010407})^{19}

\frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 7^{0} (- 5 \frac{x}{96889010407})^{20}

간소화하다 :

\frac{x ^{4}}{9688901040 7 ^{4}}

= 7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - \frac{7 ^{15} \cdot 15504 x}{96889010407} + 7^{14} \cdot 38760 (5 \frac{x}{96889010407})^{6} - 7^{13} \cdot 77520 (5 \frac{x}{96889010407})^{7} + 7^{12} \cdot 125970 (5 \frac{x}{96889010407})^{8} - 7^{11} \cdot 167960 (5 \frac{x}{96889010407})^{9} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - 6777791831720 (5 \frac{x}{96889010407})^{11} + 726191981970 (5 \frac{x}{96889010407})^{12} - 63841053360 (5 \frac{x}{96889010407})^{13} + 4560075240 (5 \frac{x}{96889010407})^{14} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + 11632845 (5 \frac{x}{96889010407})^{16} - 391020 (5 \frac{x}{96889010407})^{17} + 9310 (5 \frac{x}{96889010407})^{18} - 140 (5 \frac{x}{96889010407})^{19} + \frac{x ^{4}}{9688901040 7 ^{4}}

7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - \frac{7 ^{15} \cdot 15504 x}{96889010407} + 7^{14} \cdot 38760 (5 \frac{x}{96889010407})^{6} - 7^{13} \cdot 77520 (5 \frac{x}{96889010407})^{7} + 7^{12} \cdot 125970 (5 \frac{x}{96889010407})^{8} - 7^{11} \cdot 167960 (5 \frac{x}{96889010407})^{9} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - 6777791831720 (5 \frac{x}{96889010407})^{11} + 726191981970 (5 \frac{x}{96889010407})^{12} - 63841053360 (5 \frac{x}{96889010407})^{13} + 4560075240 (5 \frac{x}{96889010407})^{14} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + 11632845 (5 \frac{x}{96889010407})^{16} - 391020 (5 \frac{x}{96889010407})^{17} + 9310 (5 \frac{x}{96889010407})^{18} - 140 (5 \frac{x}{96889010407})^{19} + \frac{x ^{4}}{9688901040 7 ^{4}}

간소화하다 :

7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - 759696 x + 271320 x 5 \frac{x}{96889010407} - \frac{77520 x 5 x ^{2}}{5 9688901040 7 ^{2}} + \frac{125970 x 5 x ^{3}}{7 5 9688901040 7 ^{3}} - \frac{167960 x 5 x ^{4}}{49 5 9688901040 7 ^{4}} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - \frac{403271960 \cdot 240 1 ^{\frac{3}{5}} x ^{\frac{11}{5}}}{9688901040 7 ^{2}} + \frac{726191981970 x ^{2} 5 x ^{2}}{9688901040 7 ^{2} 5 9688901040 7 ^{2}} - \frac{63841053360 x ^{2} 5 x ^{3}}{9688901040 7 ^{2} 5 9688901040 7 ^{3}} + \frac{4560075240 x ^{2} 5 x ^{4}}{9688901040 7 ^{2} 5 9688901040 7 ^{4}} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + \frac{33915 5 49 x ^{\frac{16}{5}}}{9688901040 7 ^{3}} - \frac{391020 x ^{3} 5 x ^{2}}{9688901040 7 ^{3} 5 9688901040 7 ^{2}} + \frac{9310 x ^{3} 5 x ^{3}}{9688901040 7 ^{3} 5 9688901040 7 ^{3}} - \frac{140 x ^{3} 5 x ^{4}}{9688901040 7 ^{3} 5 9688901040 7 ^{4}} + \frac{x ^{4}}{9688901040 7 ^{4}}

= 7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - 759696 x + 271320 x 5 \frac{x}{96889010407} - \frac{77520 x 5 x ^{2}}{5 9688901040 7 ^{2}} + \frac{125970 x 5 x ^{3}}{7 5 9688901040 7 ^{3}} - \frac{167960 x 5 x ^{4}}{49 5 9688901040 7 ^{4}} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - \frac{403271960 \cdot 240 1 ^{\frac{3}{5}} x ^{\frac{11}{5}}}{9688901040 7 ^{2}} + \frac{726191981970 x ^{2} 5 x ^{2}}{9688901040 7 ^{2} 5 9688901040 7 ^{2}} - \frac{63841053360 x ^{2} 5 x ^{3}}{9688901040 7 ^{2} 5 9688901040 7 ^{3}} + \frac{4560075240 x ^{2} 5 x ^{4}}{9688901040 7 ^{2} 5 9688901040 7 ^{4}} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + \frac{33915 5 49 x ^{\frac{16}{5}}}{9688901040 7 ^{3}} - \frac{391020 x ^{3} 5 x ^{2}}{9688901040 7 ^{3} 5 9688901040 7 ^{2}} + \frac{9310 x ^{3} 5 x ^{3}}{9688901040 7 ^{3} 5 9688901040 7 ^{3}} - \frac{140 x ^{3} 5 x ^{4}}{9688901040 7 ^{3} 5 9688901040 7 ^{4}} + \frac{x ^{4}}{9688901040 7 ^{4}}

7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - 759696 x + 271320 x 5 \frac{x}{96889010407} - \frac{77520 x 5 x ^{2}}{5 9688901040 7 ^{2}} + \frac{125970 x 5 x ^{3}}{7 5 9688901040 7 ^{3}} - \frac{167960 x 5 x ^{4}}{49 5 9688901040 7 ^{4}} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - \frac{403271960 \cdot 240 1 ^{\frac{3}{5}} x ^{\frac{11}{5}}}{9688901040 7 ^{2}} + \frac{726191981970 x ^{2} 5 x ^{2}}{9688901040 7 ^{2} 5 9688901040 7 ^{2}} - \frac{63841053360 x ^{2} 5 x ^{3}}{9688901040 7 ^{2} 5 9688901040 7 ^{3}} + \frac{4560075240 x ^{2} 5 x ^{4}}{9688901040 7 ^{2} 5 9688901040 7 ^{4}} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + \frac{33915 5 49 x ^{\frac{16}{5}}}{9688901040 7 ^{3}} - \frac{391020 x ^{3} 5 x ^{2}}{9688901040 7 ^{3} 5 9688901040 7 ^{2}} + \frac{9310 x ^{3} 5 x ^{3}}{9688901040 7 ^{3} 5 9688901040 7 ^{3}} - \frac{140 x ^{3} 5 x ^{4}}{9688901040 7 ^{3} 5 9688901040 7 ^{4}} + \frac{x ^{4}}{9688901040 7 ^{4}}

간소화하다 :

7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - 759696 x + 271320 x 5 \frac{x}{96889010407} - \frac{77520 x 5 x ^{2}}{5 9688901040 7 ^{2}} + \frac{125970 x 5 x ^{3}}{7 5 9688901040 7 ^{3}} - \frac{167960 x 5 x ^{4}}{49 5 9688901040 7 ^{4}} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - \frac{403271960 5 240 1 ^{3} 5 x ^{11}}{9688901040 7 ^{2}} + \frac{726191981970 x ^{2} 5 x ^{2}}{9688901040 7 ^{2} 5 9688901040 7 ^{2}} - \frac{63841053360 x ^{2} 5 x ^{3}}{9688901040 7 ^{2} 5 9688901040 7 ^{3}} + \frac{4560075240 x ^{2} 5 x ^{4}}{9688901040 7 ^{2} 5 9688901040 7 ^{4}} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + \frac{33915 5 49 5 x ^{16}}{9688901040 7 ^{3}} - \frac{391020 x ^{3} 5 x ^{2}}{9688901040 7 ^{3} 5 9688901040 7 ^{2}} + \frac{9310 x ^{3} 5 x ^{3}}{9688901040 7 ^{3} 5 9688901040 7 ^{3}} - \frac{140 x ^{3} 5 x ^{4}}{9688901040 7 ^{3} 5 9688901040 7 ^{4}} + \frac{x ^{4}}{9688901040 7 ^{4}}

= 7^{20} - \frac{7 ^{19} \cdot 20 5 x}{5 96889010407} + 7^{18} \cdot 190 (5 \frac{x}{96889010407})^{2} - 7^{17} \cdot 1140 (5 \frac{x}{96889010407})^{3} + 7^{16} \cdot 4845 (5 \frac{x}{96889010407})^{4} - 759696 x + 271320 x 5 \frac{x}{96889010407} - \frac{77520 x 5 x ^{2}}{5 9688901040 7 ^{2}} + \frac{125970 x 5 x ^{3}}{7 5 9688901040 7 ^{3}} - \frac{167960 x 5 x ^{4}}{49 5 9688901040 7 ^{4}} + \frac{1.89383 E 20 x ^{2}}{9688901040 7 ^{2} \cdot 3628800} - \frac{403271960 5 240 1 ^{3} 5 x ^{11}}{9688901040 7 ^{2}} + \frac{726191981970 x ^{2} 5 x ^{2}}{9688901040 7 ^{2} 5 9688901040 7 ^{2}} - \frac{63841053360 x ^{2} 5 x ^{3}}{9688901040 7 ^{2} 5 9688901040 7 ^{3}} + \frac{4560075240 x ^{2} 5 x ^{4}}{9688901040 7 ^{2} 5 9688901040 7 ^{4}} - \frac{260575728 x ^{3}}{9688901040 7 ^{3}} + \frac{33915 5 49 5 x ^{16}}{9688901040 7 ^{3}} - \frac{391020 x ^{3} 5 x ^{2}}{9688901040 7 ^{3} 5 9688901040 7 ^{2}} + \frac{9310 x ^{3} 5 x ^{3}}{9688901040 7 ^{3} 5 9688901040 7 ^{3}} - \frac{140 x ^{3} 5 x ^{4}}{9688901040 7 ^{3} 5 9688901040 7 ^{4}} + \frac{x ^{4}}{9688901040 7 ^{4}}

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