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확장 (6-\sqrt[4]{x/(78364164096)})^{20}

해법

확장

(6 - 4 \frac{x}{78364164096})^{20}

해법

6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + 174420 x - \frac{646 2 4 x ^{5}}{3 3} + \frac{1615 x ^{3}}{136048896} - \frac{12920 x 4 x ^{3}}{4 7836416409 6 ^{3}} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{6.09354 E 13 4 x ^{9}}{4 7836416409 6 ^{9}} + \frac{1.11715 E 13 x ^{5}}{7836416409 6 ^{5}} - \frac{1692649820160 x ^{2} 4 x ^{3}}{7836416409 6 ^{2} 4 7836416409 6 ^{3}} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - \frac{4186080 4 4 8 ^{2} 2 4 x ^{13}}{7836416409 6 ^{3}} + \frac{6460 x ^{7}}{7836416409 6 ^{3}} - \frac{120559104 x ^{3} 4 x ^{3}}{7836416409 6 ^{3} 4 7836416409 6 ^{3}} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - \frac{190 6 4 x ^{17}}{7836416409 6 ^{4}} + \frac{95 x ^{9}}{7836416409 6 ^{4} \cdot 3888} - \frac{120 x ^{4} 4 x ^{3}}{7836416409 6 ^{4} 4 7836416409 6 ^{3}} + \frac{x ^{5}}{7836416409 6 ^{5}}

솔루션 단계

(6 - 4 \frac{x}{78364164096})^{20}

이항 정리 적용:

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

a = 6, b = - 4 \frac{x}{78364164096}

= i = 0 \sum 20 (i 20) \cdot 6^{(20 - i)} (- 4 \frac{x}{78364164096})^{i}

합계 확장

= \frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 6^{20} (- 4 \frac{x}{78364164096})^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 6^{19} (- 4 \frac{x}{78364164096})^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 6^{18} (- 4 \frac{x}{78364164096})^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 6^{17} (- 4 \frac{x}{78364164096})^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 6^{16} (- 4 \frac{x}{78364164096})^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 6^{15} (- 4 \frac{x}{78364164096})^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 6^{14} (- 4 \frac{x}{78364164096})^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 6^{13} (- 4 \frac{x}{78364164096})^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 6^{12} (- 4 \frac{x}{78364164096})^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 6^{11} (- 4 \frac{x}{78364164096})^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 6^{10} (- 4 \frac{x}{78364164096})^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 6^{9} (- 4 \frac{x}{78364164096})^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 6^{8} (- 4 \frac{x}{78364164096})^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 6^{7} (- 4 \frac{x}{78364164096})^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 6^{6} (- 4 \frac{x}{78364164096})^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 6^{5} (- 4 \frac{x}{78364164096})^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 6^{4} (- 4 \frac{x}{78364164096})^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 6^{3} (- 4 \frac{x}{78364164096})^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 6^{2} (- 4 \frac{x}{78364164096})^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 6^{1} (- 4 \frac{x}{78364164096})^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 6^{0} (- 4 \frac{x}{78364164096})^{20}

\frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 6^{20} (- 4 \frac{x}{78364164096})^{0}

간소화하다 :

6^{20}

\frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 6^{19} (- 4 \frac{x}{78364164096})^{1}

간소화하다 :

- \frac{6 ^{19} \cdot 20 4 x}{4 78364164096}

\frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 6^{18} (- 4 \frac{x}{78364164096})^{2}

간소화하다 :

6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2}

\frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 6^{17} (- 4 \frac{x}{78364164096})^{3}

간소화하다 :

- 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3}

\frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 6^{16} (- 4 \frac{x}{78364164096})^{4}

간소화하다 :

\frac{6 ^{16} \cdot 1615 x}{26121388032}

\frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 6^{15} (- 4 \frac{x}{78364164096})^{5}

간소화하다 :

- 6^{15} \cdot 15504 (4 \frac{x}{78364164096})^{5}

\frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 6^{14} (- 4 \frac{x}{78364164096})^{6}

간소화하다 :

6^{14} \cdot 38760 (4 \frac{x}{78364164096})^{6}

\frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 6^{13} (- 4 \frac{x}{78364164096})^{7}

간소화하다 :

- 6^{13} \cdot 77520 (4 \frac{x}{78364164096})^{7}

\frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 6^{12} (- 4 \frac{x}{78364164096})^{8}

간소화하다 :

\frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}}

\frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 6^{11} (- 4 \frac{x}{78364164096})^{9}

간소화하다 :

- \frac{2.21122 E 19 ( 4 x ) ^{9}}{362880 \cdot 7836416409 6 ^{\frac{9}{4}}}

\frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 6^{10} (- 4 \frac{x}{78364164096})^{10}

간소화하다 :

\frac{4.05391 E 19 ( 4 x ) ^{10}}{3628800 \cdot 7836416409 6 ^{\frac{5}{2}}}

\frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 6^{9} (- 4 \frac{x}{78364164096})^{11}

간소화하다 :

- 1692649820160 (4 \frac{x}{78364164096})^{11}

\frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 6^{8} (- 4 \frac{x}{78364164096})^{12}

간소화하다 :

\frac{211581227520 x ^{3}}{7836416409 6 ^{3}}

\frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 6^{7} (- 4 \frac{x}{78364164096})^{13}

간소화하다 :

- 21700638720 (4 \frac{x}{78364164096})^{13}

\frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 6^{6} (- 4 \frac{x}{78364164096})^{14}

간소화하다 :

1808386560 (4 \frac{x}{78364164096})^{14}

\frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 6^{5} (- 4 \frac{x}{78364164096})^{15}

간소화하다 :

- 120559104 (4 \frac{x}{78364164096})^{15}

\frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 6^{4} (- 4 \frac{x}{78364164096})^{16}

간소화하다 :

\frac{6279120 x ^{4}}{7836416409 6 ^{4}}

\frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 6^{3} (- 4 \frac{x}{78364164096})^{17}

간소화하다 :

- 246240 (4 \frac{x}{78364164096})^{17}

\frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 6^{2} (- 4 \frac{x}{78364164096})^{18}

간소화하다 :

6840 (4 \frac{x}{78364164096})^{18}

\frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 6^{1} (- 4 \frac{x}{78364164096})^{19}

간소화하다 :

- 120 (4 \frac{x}{78364164096})^{19}

\frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 6^{0} (- 4 \frac{x}{78364164096})^{20}

간소화하다 :

\frac{x ^{5}}{7836416409 6 ^{5}}

= 6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + \frac{6 ^{16} \cdot 1615 x}{26121388032} - 6^{15} \cdot 15504 (4 \frac{x}{78364164096})^{5} + 6^{14} \cdot 38760 (4 \frac{x}{78364164096})^{6} - 6^{13} \cdot 77520 (4 \frac{x}{78364164096})^{7} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{2.21122 E 19 ( 4 x ) ^{9}}{362880 \cdot 7836416409 6 ^{\frac{9}{4}}} + \frac{4.05391 E 19 ( 4 x ) ^{10}}{3628800 \cdot 7836416409 6 ^{\frac{5}{2}}} - 1692649820160 (4 \frac{x}{78364164096})^{11} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - 21700638720 (4 \frac{x}{78364164096})^{13} + 1808386560 (4 \frac{x}{78364164096})^{14} - 120559104 (4 \frac{x}{78364164096})^{15} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - 246240 (4 \frac{x}{78364164096})^{17} + 6840 (4 \frac{x}{78364164096})^{18} - 120 (4 \frac{x}{78364164096})^{19} + \frac{x ^{5}}{7836416409 6 ^{5}}

6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + \frac{6 ^{16} \cdot 1615 x}{26121388032} - 6^{15} \cdot 15504 (4 \frac{x}{78364164096})^{5} + 6^{14} \cdot 38760 (4 \frac{x}{78364164096})^{6} - 6^{13} \cdot 77520 (4 \frac{x}{78364164096})^{7} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{2.21122 E 19 ( 4 x ) ^{9}}{362880 \cdot 7836416409 6 ^{\frac{9}{4}}} + \frac{4.05391 E 19 ( 4 x ) ^{10}}{3628800 \cdot 7836416409 6 ^{\frac{5}{2}}} - 1692649820160 (4 \frac{x}{78364164096})^{11} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - 21700638720 (4 \frac{x}{78364164096})^{13} + 1808386560 (4 \frac{x}{78364164096})^{14} - 120559104 (4 \frac{x}{78364164096})^{15} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - 246240 (4 \frac{x}{78364164096})^{17} + 6840 (4 \frac{x}{78364164096})^{18} - 120 (4 \frac{x}{78364164096})^{19} + \frac{x ^{5}}{7836416409 6 ^{5}}

간소화하다 :

6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + 174420 x - \frac{646 2 x ^{\frac{5}{4}}}{3 3} + \frac{1615 x ^{\frac{3}{2}}}{136048896} - \frac{12920 x 4 x ^{3}}{4 7836416409 6 ^{3}} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{6.09354 E 13 x ^{\frac{9}{4}}}{7836416409 6 ^{\frac{9}{4}}} + \frac{1.11715 E 13 x ^{\frac{5}{2}}}{7836416409 6 ^{\frac{5}{2}}} - \frac{1692649820160 x ^{2} 4 x ^{3}}{7836416409 6 ^{2} 4 7836416409 6 ^{3}} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - \frac{4186080 4 4 8 ^{2} 2 x ^{\frac{13}{4}}}{7836416409 6 ^{3}} + \frac{6460 x ^{\frac{7}{2}}}{7836416409 6 ^{3}} - \frac{120559104 x ^{3} 4 x ^{3}}{7836416409 6 ^{3} 4 7836416409 6 ^{3}} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - \frac{190 6 x ^{\frac{17}{4}}}{7836416409 6 ^{4}} + \frac{95 x ^{\frac{9}{2}}}{7836416409 6 ^{4} \cdot 3888} - \frac{120 x ^{4} 4 x ^{3}}{7836416409 6 ^{4} 4 7836416409 6 ^{3}} + \frac{x ^{5}}{7836416409 6 ^{5}}

= 6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + 174420 x - \frac{646 2 x ^{\frac{5}{4}}}{3 3} + \frac{1615 x ^{\frac{3}{2}}}{136048896} - \frac{12920 x 4 x ^{3}}{4 7836416409 6 ^{3}} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{6.09354 E 13 x ^{\frac{9}{4}}}{7836416409 6 ^{\frac{9}{4}}} + \frac{1.11715 E 13 x ^{\frac{5}{2}}}{7836416409 6 ^{\frac{5}{2}}} - \frac{1692649820160 x ^{2} 4 x ^{3}}{7836416409 6 ^{2} 4 7836416409 6 ^{3}} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - \frac{4186080 4 4 8 ^{2} 2 x ^{\frac{13}{4}}}{7836416409 6 ^{3}} + \frac{6460 x ^{\frac{7}{2}}}{7836416409 6 ^{3}} - \frac{120559104 x ^{3} 4 x ^{3}}{7836416409 6 ^{3} 4 7836416409 6 ^{3}} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - \frac{190 6 x ^{\frac{17}{4}}}{7836416409 6 ^{4}} + \frac{95 x ^{\frac{9}{2}}}{7836416409 6 ^{4} \cdot 3888} - \frac{120 x ^{4} 4 x ^{3}}{7836416409 6 ^{4} 4 7836416409 6 ^{3}} + \frac{x ^{5}}{7836416409 6 ^{5}}

6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + 174420 x - \frac{646 2 x ^{\frac{5}{4}}}{3 3} + \frac{1615 x ^{\frac{3}{2}}}{136048896} - \frac{12920 x 4 x ^{3}}{4 7836416409 6 ^{3}} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{6.09354 E 13 x ^{\frac{9}{4}}}{7836416409 6 ^{\frac{9}{4}}} + \frac{1.11715 E 13 x ^{\frac{5}{2}}}{7836416409 6 ^{\frac{5}{2}}} - \frac{1692649820160 x ^{2} 4 x ^{3}}{7836416409 6 ^{2} 4 7836416409 6 ^{3}} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - \frac{4186080 4 4 8 ^{2} 2 x ^{\frac{13}{4}}}{7836416409 6 ^{3}} + \frac{6460 x ^{\frac{7}{2}}}{7836416409 6 ^{3}} - \frac{120559104 x ^{3} 4 x ^{3}}{7836416409 6 ^{3} 4 7836416409 6 ^{3}} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - \frac{190 6 x ^{\frac{17}{4}}}{7836416409 6 ^{4}} + \frac{95 x ^{\frac{9}{2}}}{7836416409 6 ^{4} \cdot 3888} - \frac{120 x ^{4} 4 x ^{3}}{7836416409 6 ^{4} 4 7836416409 6 ^{3}} + \frac{x ^{5}}{7836416409 6 ^{5}}

간소화하다 :

6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + 174420 x - \frac{646 2 4 x ^{5}}{3 3} + \frac{1615 x ^{3}}{136048896} - \frac{12920 x 4 x ^{3}}{4 7836416409 6 ^{3}} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{6.09354 E 13 4 x ^{9}}{4 7836416409 6 ^{9}} + \frac{1.11715 E 13 x ^{5}}{7836416409 6 ^{5}} - \frac{1692649820160 x ^{2} 4 x ^{3}}{7836416409 6 ^{2} 4 7836416409 6 ^{3}} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - \frac{4186080 4 4 8 ^{2} 2 4 x ^{13}}{7836416409 6 ^{3}} + \frac{6460 x ^{7}}{7836416409 6 ^{3}} - \frac{120559104 x ^{3} 4 x ^{3}}{7836416409 6 ^{3} 4 7836416409 6 ^{3}} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - \frac{190 6 4 x ^{17}}{7836416409 6 ^{4}} + \frac{95 x ^{9}}{7836416409 6 ^{4} \cdot 3888} - \frac{120 x ^{4} 4 x ^{3}}{7836416409 6 ^{4} 4 7836416409 6 ^{3}} + \frac{x ^{5}}{7836416409 6 ^{5}}

= 6^{20} - \frac{6 ^{19} \cdot 20 4 x}{4 78364164096} + 6^{18} \cdot 190 (4 \frac{x}{78364164096})^{2} - 6^{17} \cdot 1140 (4 \frac{x}{78364164096})^{3} + 174420 x - \frac{646 2 4 x ^{5}}{3 3} + \frac{1615 x ^{3}}{136048896} - \frac{12920 x 4 x ^{3}}{4 7836416409 6 ^{3}} + \frac{6 ^{12} \cdot 125970 x ^{2}}{7836416409 6 ^{2}} - \frac{6.09354 E 13 4 x ^{9}}{4 7836416409 6 ^{9}} + \frac{1.11715 E 13 x ^{5}}{7836416409 6 ^{5}} - \frac{1692649820160 x ^{2} 4 x ^{3}}{7836416409 6 ^{2} 4 7836416409 6 ^{3}} + \frac{211581227520 x ^{3}}{7836416409 6 ^{3}} - \frac{4186080 4 4 8 ^{2} 2 4 x ^{13}}{7836416409 6 ^{3}} + \frac{6460 x ^{7}}{7836416409 6 ^{3}} - \frac{120559104 x ^{3} 4 x ^{3}}{7836416409 6 ^{3} 4 7836416409 6 ^{3}} + \frac{6279120 x ^{4}}{7836416409 6 ^{4}} - \frac{190 6 4 x ^{17}}{7836416409 6 ^{4}} + \frac{95 x ^{9}}{7836416409 6 ^{4} \cdot 3888} - \frac{120 x ^{4} 4 x ^{3}}{7836416409 6 ^{4} 4 7836416409 6 ^{3}} + \frac{x ^{5}}{7836416409 6 ^{5}}

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