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인기 있는 대수학 >

확장 (5-\sqrt[3]{x/(244140625)})^{17}

해법

확장

(5 - 3 \frac{x}{244140625})^{17}

해법

5^{17} - 4150390625 3 x + 5^{15} \cdot 136 (3 \frac{x}{244140625})^{2} - 17000 x + 11900 x 3 \frac{x}{244140625} - \frac{6188 x 3 x ^{2}}{3 24414062 5 ^{2}} + \frac{12376 x ^{2}}{1220703125} - \frac{19448 x ^{2} 3 x}{6103515625 3 244140625} + \frac{4862 x ^{2} 3 x ^{2}}{6103515625 3 24414062 5 ^{2}} - \frac{4862 x ^{3}}{5 ^{27}} + \frac{19448 x ^{3} 3 x}{7812 5 ^{2} \cdot 312 5 ^{3} 3 244140625} - \frac{12376 x ^{3} 3 x ^{2}}{1562 5 ^{5} 3 24414062 5 ^{2}} + \frac{6188 x ^{4}}{5 ^{43}} - \frac{476 x ^{4} 3 x}{312 5 ^{7} \cdot 2 5 ^{4} 3 244140625} + \frac{136 x ^{4} 3 x ^{2}}{62 5 ^{11} 3 24414062 5 ^{2}} - \frac{136 x ^{5}}{2 5 ^{29}} + \frac{17 x ^{5} 3 x}{5 ^{59} 3 244140625} - \frac{x ^{5} ( 3 \frac{x}{244140625} ) ^{2}}{24414062 5 ^{5}}

솔루션 단계

(5 - 3 \frac{x}{244140625})^{17}

이항 정리 적용:

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

a = 5, b = - 3 \frac{x}{244140625}

= i = 0 \sum 17 (i 17) \cdot 5^{(17 - i)} (- 3 \frac{x}{244140625})^{i}

합계 확장

= \frac{17 !}{0 ! ( 17 - 0 ) !} \cdot 5^{17} (- 3 \frac{x}{244140625})^{0} + \frac{17 !}{1 ! ( 17 - 1 ) !} \cdot 5^{16} (- 3 \frac{x}{244140625})^{1} + \frac{17 !}{2 ! ( 17 - 2 ) !} \cdot 5^{15} (- 3 \frac{x}{244140625})^{2} + \frac{17 !}{3 ! ( 17 - 3 ) !} \cdot 5^{14} (- 3 \frac{x}{244140625})^{3} + \frac{17 !}{4 ! ( 17 - 4 ) !} \cdot 5^{13} (- 3 \frac{x}{244140625})^{4} + \frac{17 !}{5 ! ( 17 - 5 ) !} \cdot 5^{12} (- 3 \frac{x}{244140625})^{5} + \frac{17 !}{6 ! ( 17 - 6 ) !} \cdot 5^{11} (- 3 \frac{x}{244140625})^{6} + \frac{17 !}{7 ! ( 17 - 7 ) !} \cdot 5^{10} (- 3 \frac{x}{244140625})^{7} + \frac{17 !}{8 ! ( 17 - 8 ) !} \cdot 5^{9} (- 3 \frac{x}{244140625})^{8} + \frac{17 !}{9 ! ( 17 - 9 ) !} \cdot 5^{8} (- 3 \frac{x}{244140625})^{9} + \frac{17 !}{10 ! ( 17 - 10 ) !} \cdot 5^{7} (- 3 \frac{x}{244140625})^{10} + \frac{17 !}{11 ! ( 17 - 11 ) !} \cdot 5^{6} (- 3 \frac{x}{244140625})^{11} + \frac{17 !}{12 ! ( 17 - 12 ) !} \cdot 5^{5} (- 3 \frac{x}{244140625})^{12} + \frac{17 !}{13 ! ( 17 - 13 ) !} \cdot 5^{4} (- 3 \frac{x}{244140625})^{13} + \frac{17 !}{14 ! ( 17 - 14 ) !} \cdot 5^{3} (- 3 \frac{x}{244140625})^{14} + \frac{17 !}{15 ! ( 17 - 15 ) !} \cdot 5^{2} (- 3 \frac{x}{244140625})^{15} + \frac{17 !}{16 ! ( 17 - 16 ) !} \cdot 5^{1} (- 3 \frac{x}{244140625})^{16} + \frac{17 !}{17 ! ( 17 - 17 ) !} \cdot 5^{0} (- 3 \frac{x}{244140625})^{17}

\frac{17 !}{0 ! ( 17 - 0 ) !} \cdot 5^{17} (- 3 \frac{x}{244140625})^{0}

간소화하다 :

5^{17}

\frac{17 !}{1 ! ( 17 - 1 ) !} \cdot 5^{16} (- 3 \frac{x}{244140625})^{1}

간소화하다 :

- \frac{5 ^{16} \cdot 17 3 x}{3 244140625}

\frac{17 !}{2 ! ( 17 - 2 ) !} \cdot 5^{15} (- 3 \frac{x}{244140625})^{2}

간소화하다 :

5^{15} \cdot 136 (3 \frac{x}{244140625})^{2}

\frac{17 !}{3 ! ( 17 - 3 ) !} \cdot 5^{14} (- 3 \frac{x}{244140625})^{3}

간소화하다 :

- 17000 x

\frac{17 !}{4 ! ( 17 - 4 ) !} \cdot 5^{13} (- 3 \frac{x}{244140625})^{4}

간소화하다 :

5^{13} \cdot 2380 (3 \frac{x}{244140625})^{4}

\frac{17 !}{5 ! ( 17 - 5 ) !} \cdot 5^{12} (- 3 \frac{x}{244140625})^{5}

간소화하다 :

- 1510742187500 (3 \frac{x}{244140625})^{5}

\frac{17 !}{6 ! ( 17 - 6 ) !} \cdot 5^{11} (- 3 \frac{x}{244140625})^{6}

간소화하다 :

\frac{604296875000 x ^{2}}{24414062 5 ^{2}}

\frac{17 !}{7 ! ( 17 - 7 ) !} \cdot 5^{10} (- 3 \frac{x}{244140625})^{7}

간소화하다 :

- 189921875000 (3 \frac{x}{244140625})^{7}

\frac{17 !}{8 ! ( 17 - 8 ) !} \cdot 5^{9} (- 3 \frac{x}{244140625})^{8}

간소화하다 :

47480468750 (3 \frac{x}{244140625})^{8}

\frac{17 !}{9 ! ( 17 - 9 ) !} \cdot 5^{8} (- 3 \frac{x}{244140625})^{9}

간소화하다 :

- \frac{9496093750 x ^{3}}{24414062 5 ^{3}}

\frac{17 !}{10 ! ( 17 - 10 ) !} \cdot 5^{7} (- 3 \frac{x}{244140625})^{10}

간소화하다 :

1519375000 (3 \frac{x}{244140625})^{10}

\frac{17 !}{11 ! ( 17 - 11 ) !} \cdot 5^{6} (- 3 \frac{x}{244140625})^{11}

간소화하다 :

- 193375000 (3 \frac{x}{244140625})^{11}

\frac{17 !}{12 ! ( 17 - 12 ) !} \cdot 5^{5} (- 3 \frac{x}{244140625})^{12}

간소화하다 :

\frac{19337500 x ^{4}}{24414062 5 ^{4}}

\frac{17 !}{13 ! ( 17 - 13 ) !} \cdot 5^{4} (- 3 \frac{x}{244140625})^{13}

간소화하다 :

- 1487500 (3 \frac{x}{244140625})^{13}

\frac{17 !}{14 ! ( 17 - 14 ) !} \cdot 5^{3} (- 3 \frac{x}{244140625})^{14}

간소화하다 :

85000 (3 \frac{x}{244140625})^{14}

\frac{17 !}{15 ! ( 17 - 15 ) !} \cdot 5^{2} (- 3 \frac{x}{244140625})^{15}

간소화하다 :

- \frac{3400 x ^{5}}{24414062 5 ^{5}}

\frac{17 !}{16 ! ( 17 - 16 ) !} \cdot 5^{1} (- 3 \frac{x}{244140625})^{16}

간소화하다 :

85 (3 \frac{x}{244140625})^{16}

\frac{17 !}{17 ! ( 17 - 17 ) !} \cdot 5^{0} (- 3 \frac{x}{244140625})^{17}

간소화하다 :

- (3 \frac{x}{244140625})^{17}

= 5^{17} - \frac{5 ^{16} \cdot 17 3 x}{3 244140625} + 5^{15} \cdot 136 (3 \frac{x}{244140625})^{2} - 17000 x + 5^{13} \cdot 2380 (3 \frac{x}{244140625})^{4} - 1510742187500 (3 \frac{x}{244140625})^{5} + \frac{604296875000 x ^{2}}{24414062 5 ^{2}} - 189921875000 (3 \frac{x}{244140625})^{7} + 47480468750 (3 \frac{x}{244140625})^{8} - \frac{9496093750 x ^{3}}{24414062 5 ^{3}} + 1519375000 (3 \frac{x}{244140625})^{10} - 193375000 (3 \frac{x}{244140625})^{11} + \frac{19337500 x ^{4}}{24414062 5 ^{4}} - 1487500 (3 \frac{x}{244140625})^{13} + 85000 (3 \frac{x}{244140625})^{14} - \frac{3400 x ^{5}}{24414062 5 ^{5}} + 85 (3 \frac{x}{244140625})^{16} - (3 \frac{x}{244140625})^{17}

5^{17} - \frac{5 ^{16} \cdot 17 3 x}{3 244140625} + 5^{15} \cdot 136 (3 \frac{x}{244140625})^{2} - 17000 x + 5^{13} \cdot 2380 (3 \frac{x}{244140625})^{4} - 1510742187500 (3 \frac{x}{244140625})^{5} + \frac{604296875000 x ^{2}}{24414062 5 ^{2}} - 189921875000 (3 \frac{x}{244140625})^{7} + 47480468750 (3 \frac{x}{244140625})^{8} - \frac{9496093750 x ^{3}}{24414062 5 ^{3}} + 1519375000 (3 \frac{x}{244140625})^{10} - 193375000 (3 \frac{x}{244140625})^{11} + \frac{19337500 x ^{4}}{24414062 5 ^{4}} - 1487500 (3 \frac{x}{244140625})^{13} + 85000 (3 \frac{x}{244140625})^{14} - \frac{3400 x ^{5}}{24414062 5 ^{5}} + 85 (3 \frac{x}{244140625})^{16} - (3 \frac{x}{244140625})^{17}

간소화하다 :

5^{17} - 4150390625 3 x + 5^{15} \cdot 136 (3 \frac{x}{244140625})^{2} - 17000 x + 11900 x 3 \frac{x}{244140625} - \frac{6188 x 3 x ^{2}}{3 24414062 5 ^{2}} + \frac{12376 x ^{2}}{1220703125} - \frac{19448 x ^{2} 3 x}{6103515625 3 244140625} + \frac{4862 x ^{2} 3 x ^{2}}{6103515625 3 24414062 5 ^{2}} - \frac{4862 x ^{3}}{5 ^{27}} + \frac{19448 x ^{3} 3 x}{7812 5 ^{2} \cdot 312 5 ^{3} 3 244140625} - \frac{12376 x ^{3} 3 x ^{2}}{1562 5 ^{5} 3 24414062 5 ^{2}} + \frac{6188 x ^{4}}{5 ^{43}} - \frac{476 x ^{4} 3 x}{312 5 ^{7} \cdot 2 5 ^{4} 3 244140625} + \frac{136 x ^{4} 3 x ^{2}}{62 5 ^{11} 3 24414062 5 ^{2}} - \frac{136 x ^{5}}{2 5 ^{29}} + \frac{17 x ^{5} 3 x}{5 ^{59} 3 244140625} - \frac{x ^{5} ( 3 \frac{x}{244140625} ) ^{2}}{24414062 5 ^{5}}

= 5^{17} - 4150390625 3 x + 5^{15} \cdot 136 (3 \frac{x}{244140625})^{2} - 17000 x + 11900 x 3 \frac{x}{244140625} - \frac{6188 x 3 x ^{2}}{3 24414062 5 ^{2}} + \frac{12376 x ^{2}}{1220703125} - \frac{19448 x ^{2} 3 x}{6103515625 3 244140625} + \frac{4862 x ^{2} 3 x ^{2}}{6103515625 3 24414062 5 ^{2}} - \frac{4862 x ^{3}}{5 ^{27}} + \frac{19448 x ^{3} 3 x}{7812 5 ^{2} \cdot 312 5 ^{3} 3 244140625} - \frac{12376 x ^{3} 3 x ^{2}}{1562 5 ^{5} 3 24414062 5 ^{2}} + \frac{6188 x ^{4}}{5 ^{43}} - \frac{476 x ^{4} 3 x}{312 5 ^{7} \cdot 2 5 ^{4} 3 244140625} + \frac{136 x ^{4} 3 x ^{2}}{62 5 ^{11} 3 24414062 5 ^{2}} - \frac{136 x ^{5}}{2 5 ^{29}} + \frac{17 x ^{5} 3 x}{5 ^{59} 3 244140625} - \frac{x ^{5} ( 3 \frac{x}{244140625} ) ^{2}}{24414062 5 ^{5}}

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