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פּוֹפּוּלָרִי אלגברה >

expand (6-\sqrt[5]{x/(13060694016)})^{20}

פתרון

expand

(6 - 5 \frac{x}{13060694016})^{20}

פתרון

6^{20} - 1567283281920 5 21 6^{4} 5 x + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + \frac{3230 5 2 ^{2} 5 x ^{6}}{5 3 ^{3}} - \frac{77520 x 5 x ^{2}}{5 1306069401 6 ^{2}} + \frac{20995 x 5 x ^{3}}{5 1306069401 6 ^{3}} - \frac{6.09354 E 13 5 x ^{9}}{5 1306069401 6 ^{9}} + \frac{1.11715 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{435352320 5 32 4 ^{3} 5 2 5 x ^{11}}{1306069401 6 ^{2}} + \frac{211581227520 x ^{2} 5 x ^{2}}{1306069401 6 ^{2} 5 1306069401 6 ^{2}} - \frac{21700638720 x ^{2} 5 x ^{3}}{1306069401 6 ^{2} 5 1306069401 6 ^{3}} + \frac{1808386560 x ^{2} 5 x ^{4}}{1306069401 6 ^{2} 5 1306069401 6 ^{4}} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + \frac{29070 5 36 5 x ^{16}}{1306069401 6 ^{3}} - \frac{246240 x ^{3} 5 x ^{2}}{1306069401 6 ^{3} 5 1306069401 6 ^{2}} + \frac{6840 x ^{3} 5 x ^{3}}{1306069401 6 ^{3} 5 1306069401 6 ^{3}} - \frac{120 x ^{3} 5 x ^{4}}{1306069401 6 ^{3} 5 1306069401 6 ^{4}} + \frac{x ^{4}}{1306069401 6 ^{4}}

צעדי פתרון

(6 - 5 \frac{x}{13060694016})^{20}

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

:הפעל את הבינום של ניוטון

a = 6, b = - 5 \frac{x}{13060694016}

= i = 0 \sum 20 (i 20) \cdot 6^{(20 - i)} (- 5 \frac{x}{13060694016})^{i}

הרחב את הסכום

= \frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 6^{20} (- 5 \frac{x}{13060694016})^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 6^{19} (- 5 \frac{x}{13060694016})^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 6^{18} (- 5 \frac{x}{13060694016})^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 6^{17} (- 5 \frac{x}{13060694016})^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 6^{16} (- 5 \frac{x}{13060694016})^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 6^{15} (- 5 \frac{x}{13060694016})^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 6^{14} (- 5 \frac{x}{13060694016})^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 6^{13} (- 5 \frac{x}{13060694016})^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 6^{12} (- 5 \frac{x}{13060694016})^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 6^{11} (- 5 \frac{x}{13060694016})^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 6^{10} (- 5 \frac{x}{13060694016})^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 6^{9} (- 5 \frac{x}{13060694016})^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 6^{8} (- 5 \frac{x}{13060694016})^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 6^{7} (- 5 \frac{x}{13060694016})^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 6^{6} (- 5 \frac{x}{13060694016})^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 6^{5} (- 5 \frac{x}{13060694016})^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 6^{4} (- 5 \frac{x}{13060694016})^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 6^{3} (- 5 \frac{x}{13060694016})^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 6^{2} (- 5 \frac{x}{13060694016})^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 6^{1} (- 5 \frac{x}{13060694016})^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 6^{0} (- 5 \frac{x}{13060694016})^{20}

\frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 6^{20} (- 5 \frac{x}{13060694016})^{0}

פשט את:

6^{20}

\frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 6^{19} (- 5 \frac{x}{13060694016})^{1}

פשט את:

- \frac{6 ^{19} \cdot 20 5 x}{5 13060694016}

\frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 6^{18} (- 5 \frac{x}{13060694016})^{2}

פשט את:

6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2}

\frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 6^{17} (- 5 \frac{x}{13060694016})^{3}

פשט את:

- 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3}

\frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 6^{16} (- 5 \frac{x}{13060694016})^{4}

פשט את:

6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4}

\frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 6^{15} (- 5 \frac{x}{13060694016})^{5}

פשט את:

- 558144 x

\frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 6^{14} (- 5 \frac{x}{13060694016})^{6}

פשט את:

6^{14} \cdot 38760 (5 \frac{x}{13060694016})^{6}

\frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 6^{13} (- 5 \frac{x}{13060694016})^{7}

פשט את:

- 6^{13} \cdot 77520 (5 \frac{x}{13060694016})^{7}

\frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 6^{12} (- 5 \frac{x}{13060694016})^{8}

פשט את:

6^{12} \cdot 125970 (5 \frac{x}{13060694016})^{8}

\frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 6^{11} (- 5 \frac{x}{13060694016})^{9}

פשט את:

- \frac{2.21122 E 19 ( 5 x ) ^{9}}{362880 \cdot 1306069401 6 ^{\frac{9}{5}}}

\frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 6^{10} (- 5 \frac{x}{13060694016})^{10}

פשט את:

\frac{4.05391 E 19 x ^{2}}{1306069401 6 ^{2} \cdot 3628800}

\frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 6^{9} (- 5 \frac{x}{13060694016})^{11}

פשט את:

- 1692649820160 (5 \frac{x}{13060694016})^{11}

\frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 6^{8} (- 5 \frac{x}{13060694016})^{12}

פשט את:

211581227520 (5 \frac{x}{13060694016})^{12}

\frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 6^{7} (- 5 \frac{x}{13060694016})^{13}

פשט את:

- 21700638720 (5 \frac{x}{13060694016})^{13}

\frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 6^{6} (- 5 \frac{x}{13060694016})^{14}

פשט את:

1808386560 (5 \frac{x}{13060694016})^{14}

\frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 6^{5} (- 5 \frac{x}{13060694016})^{15}

פשט את:

- \frac{120559104 x ^{3}}{1306069401 6 ^{3}}

\frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 6^{4} (- 5 \frac{x}{13060694016})^{16}

פשט את:

6279120 (5 \frac{x}{13060694016})^{16}

\frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 6^{3} (- 5 \frac{x}{13060694016})^{17}

פשט את:

- 246240 (5 \frac{x}{13060694016})^{17}

\frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 6^{2} (- 5 \frac{x}{13060694016})^{18}

פשט את:

6840 (5 \frac{x}{13060694016})^{18}

\frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 6^{1} (- 5 \frac{x}{13060694016})^{19}

פשט את:

- 120 (5 \frac{x}{13060694016})^{19}

\frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 6^{0} (- 5 \frac{x}{13060694016})^{20}

פשט את:

\frac{x ^{4}}{1306069401 6 ^{4}}

= 6^{20} - \frac{6 ^{19} \cdot 20 5 x}{5 13060694016} + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + 6^{14} \cdot 38760 (5 \frac{x}{13060694016})^{6} - 6^{13} \cdot 77520 (5 \frac{x}{13060694016})^{7} + 6^{12} \cdot 125970 (5 \frac{x}{13060694016})^{8} - \frac{2.21122 E 19 ( 5 x ) ^{9}}{362880 \cdot 1306069401 6 ^{\frac{9}{5}}} + \frac{4.05391 E 19 x ^{2}}{1306069401 6 ^{2} \cdot 3628800} - 1692649820160 (5 \frac{x}{13060694016})^{11} + 211581227520 (5 \frac{x}{13060694016})^{12} - 21700638720 (5 \frac{x}{13060694016})^{13} + 1808386560 (5 \frac{x}{13060694016})^{14} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + 6279120 (5 \frac{x}{13060694016})^{16} - 246240 (5 \frac{x}{13060694016})^{17} + 6840 (5 \frac{x}{13060694016})^{18} - 120 (5 \frac{x}{13060694016})^{19} + \frac{x ^{4}}{1306069401 6 ^{4}}

6^{20} - \frac{6 ^{19} \cdot 20 5 x}{5 13060694016} + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + 6^{14} \cdot 38760 (5 \frac{x}{13060694016})^{6} - 6^{13} \cdot 77520 (5 \frac{x}{13060694016})^{7} + 6^{12} \cdot 125970 (5 \frac{x}{13060694016})^{8} - \frac{2.21122 E 19 ( 5 x ) ^{9}}{362880 \cdot 1306069401 6 ^{\frac{9}{5}}} + \frac{4.05391 E 19 x ^{2}}{1306069401 6 ^{2} \cdot 3628800} - 1692649820160 (5 \frac{x}{13060694016})^{11} + 211581227520 (5 \frac{x}{13060694016})^{12} - 21700638720 (5 \frac{x}{13060694016})^{13} + 1808386560 (5 \frac{x}{13060694016})^{14} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + 6279120 (5 \frac{x}{13060694016})^{16} - 246240 (5 \frac{x}{13060694016})^{17} + 6840 (5 \frac{x}{13060694016})^{18} - 120 (5 \frac{x}{13060694016})^{19} + \frac{x ^{4}}{1306069401 6 ^{4}}

פשט את:

6^{20} - 1567283281920 \cdot 21 6^{\frac{4}{5}} 5 x + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + \frac{3230 \cdot 2 ^{\frac{2}{5}} x ^{\frac{6}{5}}}{3 ^{\frac{3}{5}}} - \frac{77520 x 5 x ^{2}}{5 1306069401 6 ^{2}} + \frac{20995 x 5 x ^{3}}{5 1306069401 6 ^{3}} - \frac{6.09354 E 13 x ^{\frac{9}{5}}}{1306069401 6 ^{\frac{9}{5}}} + \frac{1.11715 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{435352320 \cdot 32 4 ^{\frac{3}{5}} 5 2 x ^{\frac{11}{5}}}{1306069401 6 ^{2}} + \frac{211581227520 x ^{2} 5 x ^{2}}{1306069401 6 ^{2} 5 1306069401 6 ^{2}} - \frac{21700638720 x ^{2} 5 x ^{3}}{1306069401 6 ^{2} 5 1306069401 6 ^{3}} + \frac{1808386560 x ^{2} 5 x ^{4}}{1306069401 6 ^{2} 5 1306069401 6 ^{4}} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + \frac{29070 5 36 x ^{\frac{16}{5}}}{1306069401 6 ^{3}} - \frac{246240 x ^{3} 5 x ^{2}}{1306069401 6 ^{3} 5 1306069401 6 ^{2}} + \frac{6840 x ^{3} 5 x ^{3}}{1306069401 6 ^{3} 5 1306069401 6 ^{3}} - \frac{120 x ^{3} 5 x ^{4}}{1306069401 6 ^{3} 5 1306069401 6 ^{4}} + \frac{x ^{4}}{1306069401 6 ^{4}}

= 6^{20} - 1567283281920 \cdot 21 6^{\frac{4}{5}} 5 x + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + \frac{3230 \cdot 2 ^{\frac{2}{5}} x ^{\frac{6}{5}}}{3 ^{\frac{3}{5}}} - \frac{77520 x 5 x ^{2}}{5 1306069401 6 ^{2}} + \frac{20995 x 5 x ^{3}}{5 1306069401 6 ^{3}} - \frac{6.09354 E 13 x ^{\frac{9}{5}}}{1306069401 6 ^{\frac{9}{5}}} + \frac{1.11715 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{435352320 \cdot 32 4 ^{\frac{3}{5}} 5 2 x ^{\frac{11}{5}}}{1306069401 6 ^{2}} + \frac{211581227520 x ^{2} 5 x ^{2}}{1306069401 6 ^{2} 5 1306069401 6 ^{2}} - \frac{21700638720 x ^{2} 5 x ^{3}}{1306069401 6 ^{2} 5 1306069401 6 ^{3}} + \frac{1808386560 x ^{2} 5 x ^{4}}{1306069401 6 ^{2} 5 1306069401 6 ^{4}} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + \frac{29070 5 36 x ^{\frac{16}{5}}}{1306069401 6 ^{3}} - \frac{246240 x ^{3} 5 x ^{2}}{1306069401 6 ^{3} 5 1306069401 6 ^{2}} + \frac{6840 x ^{3} 5 x ^{3}}{1306069401 6 ^{3} 5 1306069401 6 ^{3}} - \frac{120 x ^{3} 5 x ^{4}}{1306069401 6 ^{3} 5 1306069401 6 ^{4}} + \frac{x ^{4}}{1306069401 6 ^{4}}

6^{20} - 1567283281920 \cdot 21 6^{\frac{4}{5}} 5 x + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + \frac{3230 \cdot 2 ^{\frac{2}{5}} x ^{\frac{6}{5}}}{3 ^{\frac{3}{5}}} - \frac{77520 x 5 x ^{2}}{5 1306069401 6 ^{2}} + \frac{20995 x 5 x ^{3}}{5 1306069401 6 ^{3}} - \frac{6.09354 E 13 x ^{\frac{9}{5}}}{1306069401 6 ^{\frac{9}{5}}} + \frac{1.11715 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{435352320 \cdot 32 4 ^{\frac{3}{5}} 5 2 x ^{\frac{11}{5}}}{1306069401 6 ^{2}} + \frac{211581227520 x ^{2} 5 x ^{2}}{1306069401 6 ^{2} 5 1306069401 6 ^{2}} - \frac{21700638720 x ^{2} 5 x ^{3}}{1306069401 6 ^{2} 5 1306069401 6 ^{3}} + \frac{1808386560 x ^{2} 5 x ^{4}}{1306069401 6 ^{2} 5 1306069401 6 ^{4}} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + \frac{29070 5 36 x ^{\frac{16}{5}}}{1306069401 6 ^{3}} - \frac{246240 x ^{3} 5 x ^{2}}{1306069401 6 ^{3} 5 1306069401 6 ^{2}} + \frac{6840 x ^{3} 5 x ^{3}}{1306069401 6 ^{3} 5 1306069401 6 ^{3}} - \frac{120 x ^{3} 5 x ^{4}}{1306069401 6 ^{3} 5 1306069401 6 ^{4}} + \frac{x ^{4}}{1306069401 6 ^{4}}

פשט את:

6^{20} - 1567283281920 5 21 6^{4} 5 x + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + \frac{3230 5 2 ^{2} 5 x ^{6}}{5 3 ^{3}} - \frac{77520 x 5 x ^{2}}{5 1306069401 6 ^{2}} + \frac{20995 x 5 x ^{3}}{5 1306069401 6 ^{3}} - \frac{6.09354 E 13 5 x ^{9}}{5 1306069401 6 ^{9}} + \frac{1.11715 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{435352320 5 32 4 ^{3} 5 2 5 x ^{11}}{1306069401 6 ^{2}} + \frac{211581227520 x ^{2} 5 x ^{2}}{1306069401 6 ^{2} 5 1306069401 6 ^{2}} - \frac{21700638720 x ^{2} 5 x ^{3}}{1306069401 6 ^{2} 5 1306069401 6 ^{3}} + \frac{1808386560 x ^{2} 5 x ^{4}}{1306069401 6 ^{2} 5 1306069401 6 ^{4}} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + \frac{29070 5 36 5 x ^{16}}{1306069401 6 ^{3}} - \frac{246240 x ^{3} 5 x ^{2}}{1306069401 6 ^{3} 5 1306069401 6 ^{2}} + \frac{6840 x ^{3} 5 x ^{3}}{1306069401 6 ^{3} 5 1306069401 6 ^{3}} - \frac{120 x ^{3} 5 x ^{4}}{1306069401 6 ^{3} 5 1306069401 6 ^{4}} + \frac{x ^{4}}{1306069401 6 ^{4}}

= 6^{20} - 1567283281920 5 21 6^{4} 5 x + 6^{18} \cdot 190 (5 \frac{x}{13060694016})^{2} - 6^{17} \cdot 1140 (5 \frac{x}{13060694016})^{3} + 6^{16} \cdot 4845 (5 \frac{x}{13060694016})^{4} - 558144 x + \frac{3230 5 2 ^{2} 5 x ^{6}}{5 3 ^{3}} - \frac{77520 x 5 x ^{2}}{5 1306069401 6 ^{2}} + \frac{20995 x 5 x ^{3}}{5 1306069401 6 ^{3}} - \frac{6.09354 E 13 5 x ^{9}}{5 1306069401 6 ^{9}} + \frac{1.11715 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{435352320 5 32 4 ^{3} 5 2 5 x ^{11}}{1306069401 6 ^{2}} + \frac{211581227520 x ^{2} 5 x ^{2}}{1306069401 6 ^{2} 5 1306069401 6 ^{2}} - \frac{21700638720 x ^{2} 5 x ^{3}}{1306069401 6 ^{2} 5 1306069401 6 ^{3}} + \frac{1808386560 x ^{2} 5 x ^{4}}{1306069401 6 ^{2} 5 1306069401 6 ^{4}} - \frac{120559104 x ^{3}}{1306069401 6 ^{3}} + \frac{29070 5 36 5 x ^{16}}{1306069401 6 ^{3}} - \frac{246240 x ^{3} 5 x ^{2}}{1306069401 6 ^{3} 5 1306069401 6 ^{2}} + \frac{6840 x ^{3} 5 x ^{3}}{1306069401 6 ^{3} 5 1306069401 6 ^{3}} - \frac{120 x ^{3} 5 x ^{4}}{1306069401 6 ^{3} 5 1306069401 6 ^{4}} + \frac{x ^{4}}{1306069401 6 ^{4}}

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