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شائع الجبر >

expand (7-\sqrt[4]{x/(678223072849)})^{20}

الحلّ

expand

(7 - 4 \frac{x}{678223072849})^{20}

الحلّ

7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + 237405 x - \frac{15504 4 x ^{5}}{4 282475249} + \frac{38760 x ^{3}}{678223072849} - \frac{77520 x 4 x ^{3}}{7 4 67822307284 9 ^{3}} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - \frac{7 ^{11} \cdot 167960 4 x ^{9}}{67822307284 9 ^{2} 4 678223072849} + \frac{1.89383 E 20 x ^{5}}{3628800 67822307284 9 ^{5}} - \frac{6777791831720 x ^{2} 4 x ^{3}}{67822307284 9 ^{2} 4 67822307284 9 ^{3}} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - \frac{26589360 7 4 x ^{13}}{67822307284 9 ^{3}} + \frac{38760 x ^{7}}{67822307284 9 ^{3} \cdot 7} - \frac{260575728 x ^{3} 4 x ^{3}}{67822307284 9 ^{3} 4 67822307284 9 ^{3}} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - \frac{1140 4 x ^{17}}{67822307284 9 ^{4} 7} + \frac{190 x ^{9}}{67822307284 9 ^{4} 282475249} - \frac{140 x ^{4} 4 x ^{3}}{67822307284 9 ^{4} 4 67822307284 9 ^{3}} + \frac{x ^{5}}{67822307284 9 ^{5}}

خطوات الحلّ

(7 - 4 \frac{x}{678223072849})^{20}

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

:فعّل نظريّة ذات الحدّين لنيوتن

a = 7, b = - 4 \frac{x}{678223072849}

= i = 0 \sum 20 (i 20) \cdot 7^{(20 - i)} (- 4 \frac{x}{678223072849})^{i}

وسّع حاصل الجمع

= \frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 7^{20} (- 4 \frac{x}{678223072849})^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 7^{19} (- 4 \frac{x}{678223072849})^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 7^{18} (- 4 \frac{x}{678223072849})^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 7^{17} (- 4 \frac{x}{678223072849})^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 7^{16} (- 4 \frac{x}{678223072849})^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 7^{15} (- 4 \frac{x}{678223072849})^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 7^{14} (- 4 \frac{x}{678223072849})^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 7^{13} (- 4 \frac{x}{678223072849})^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 7^{12} (- 4 \frac{x}{678223072849})^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 7^{11} (- 4 \frac{x}{678223072849})^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 7^{10} (- 4 \frac{x}{678223072849})^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 7^{9} (- 4 \frac{x}{678223072849})^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 7^{8} (- 4 \frac{x}{678223072849})^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 7^{7} (- 4 \frac{x}{678223072849})^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 7^{6} (- 4 \frac{x}{678223072849})^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 7^{5} (- 4 \frac{x}{678223072849})^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 7^{4} (- 4 \frac{x}{678223072849})^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 7^{3} (- 4 \frac{x}{678223072849})^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 7^{2} (- 4 \frac{x}{678223072849})^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 7^{1} (- 4 \frac{x}{678223072849})^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 7^{0} (- 4 \frac{x}{678223072849})^{20}

\frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 7^{20} (- 4 \frac{x}{678223072849})^{0}

بسّط:

7^{20}

\frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 7^{19} (- 4 \frac{x}{678223072849})^{1}

بسّط:

- \frac{7 ^{19} \cdot 20 4 x}{4 678223072849}

\frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 7^{18} (- 4 \frac{x}{678223072849})^{2}

بسّط:

7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2}

\frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 7^{17} (- 4 \frac{x}{678223072849})^{3}

بسّط:

- 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3}

\frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 7^{16} (- 4 \frac{x}{678223072849})^{4}

بسّط:

\frac{7 ^{16} \cdot 4845 x}{678223072849}

\frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 7^{15} (- 4 \frac{x}{678223072849})^{5}

بسّط:

- 7^{15} \cdot 15504 (4 \frac{x}{678223072849})^{5}

\frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 7^{14} (- 4 \frac{x}{678223072849})^{6}

بسّط:

7^{14} \cdot 38760 (4 \frac{x}{678223072849})^{6}

\frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 7^{13} (- 4 \frac{x}{678223072849})^{7}

بسّط:

- 7^{13} \cdot 77520 (4 \frac{x}{678223072849})^{7}

\frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 7^{12} (- 4 \frac{x}{678223072849})^{8}

بسّط:

\frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}}

\frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 7^{11} (- 4 \frac{x}{678223072849})^{9}

بسّط:

- 7^{11} \cdot 167960 (4 \frac{x}{678223072849})^{9}

\frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 7^{10} (- 4 \frac{x}{678223072849})^{10}

بسّط:

\frac{1.89383 E 20 ( 4 x ) ^{10}}{3628800 \cdot 67822307284 9 ^{\frac{5}{2}}}

\frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 7^{9} (- 4 \frac{x}{678223072849})^{11}

بسّط:

- 6777791831720 (4 \frac{x}{678223072849})^{11}

\frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 7^{8} (- 4 \frac{x}{678223072849})^{12}

بسّط:

\frac{726191981970 x ^{3}}{67822307284 9 ^{3}}

\frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 7^{7} (- 4 \frac{x}{678223072849})^{13}

بسّط:

- 63841053360 (4 \frac{x}{678223072849})^{13}

\frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 7^{6} (- 4 \frac{x}{678223072849})^{14}

بسّط:

4560075240 (4 \frac{x}{678223072849})^{14}

\frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 7^{5} (- 4 \frac{x}{678223072849})^{15}

بسّط:

- 260575728 (4 \frac{x}{678223072849})^{15}

\frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 7^{4} (- 4 \frac{x}{678223072849})^{16}

بسّط:

\frac{11632845 x ^{4}}{67822307284 9 ^{4}}

\frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 7^{3} (- 4 \frac{x}{678223072849})^{17}

بسّط:

- 391020 (4 \frac{x}{678223072849})^{17}

\frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 7^{2} (- 4 \frac{x}{678223072849})^{18}

بسّط:

9310 (4 \frac{x}{678223072849})^{18}

\frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 7^{1} (- 4 \frac{x}{678223072849})^{19}

بسّط:

- 140 (4 \frac{x}{678223072849})^{19}

\frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 7^{0} (- 4 \frac{x}{678223072849})^{20}

بسّط:

\frac{x ^{5}}{67822307284 9 ^{5}}

= 7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + \frac{7 ^{16} \cdot 4845 x}{678223072849} - 7^{15} \cdot 15504 (4 \frac{x}{678223072849})^{5} + 7^{14} \cdot 38760 (4 \frac{x}{678223072849})^{6} - 7^{13} \cdot 77520 (4 \frac{x}{678223072849})^{7} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - 7^{11} \cdot 167960 (4 \frac{x}{678223072849})^{9} + \frac{1.89383 E 20 ( 4 x ) ^{10}}{3628800 \cdot 67822307284 9 ^{\frac{5}{2}}} - 6777791831720 (4 \frac{x}{678223072849})^{11} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - 63841053360 (4 \frac{x}{678223072849})^{13} + 4560075240 (4 \frac{x}{678223072849})^{14} - 260575728 (4 \frac{x}{678223072849})^{15} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - 391020 (4 \frac{x}{678223072849})^{17} + 9310 (4 \frac{x}{678223072849})^{18} - 140 (4 \frac{x}{678223072849})^{19} + \frac{x ^{5}}{67822307284 9 ^{5}}

7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + \frac{7 ^{16} \cdot 4845 x}{678223072849} - 7^{15} \cdot 15504 (4 \frac{x}{678223072849})^{5} + 7^{14} \cdot 38760 (4 \frac{x}{678223072849})^{6} - 7^{13} \cdot 77520 (4 \frac{x}{678223072849})^{7} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - 7^{11} \cdot 167960 (4 \frac{x}{678223072849})^{9} + \frac{1.89383 E 20 ( 4 x ) ^{10}}{3628800 \cdot 67822307284 9 ^{\frac{5}{2}}} - 6777791831720 (4 \frac{x}{678223072849})^{11} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - 63841053360 (4 \frac{x}{678223072849})^{13} + 4560075240 (4 \frac{x}{678223072849})^{14} - 260575728 (4 \frac{x}{678223072849})^{15} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - 391020 (4 \frac{x}{678223072849})^{17} + 9310 (4 \frac{x}{678223072849})^{18} - 140 (4 \frac{x}{678223072849})^{19} + \frac{x ^{5}}{67822307284 9 ^{5}}

بسّط:

7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + 237405 x - \frac{15504 x ^{\frac{5}{4}}}{4 282475249} + \frac{38760 x ^{\frac{3}{2}}}{678223072849} - \frac{77520 x 4 x ^{3}}{7 4 67822307284 9 ^{3}} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - \frac{7 ^{11} \cdot 167960 x ^{\frac{9}{4}}}{67822307284 9 ^{2} 4 678223072849} + \frac{1.89383 E 20 x ^{\frac{5}{2}}}{3628800 \cdot 67822307284 9 ^{\frac{5}{2}}} - \frac{6777791831720 x ^{2} 4 x ^{3}}{67822307284 9 ^{2} 4 67822307284 9 ^{3}} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - \frac{26589360 7 x ^{\frac{13}{4}}}{67822307284 9 ^{3}} + \frac{38760 x ^{\frac{7}{2}}}{67822307284 9 ^{3} \cdot 7} - \frac{260575728 x ^{3} 4 x ^{3}}{67822307284 9 ^{3} 4 67822307284 9 ^{3}} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - \frac{1140 x ^{\frac{17}{4}}}{67822307284 9 ^{4} \cdot 7 ^{\frac{1}{2}}} + \frac{190 x ^{\frac{9}{2}}}{67822307284 9 ^{4} 282475249} - \frac{140 x ^{4} 4 x ^{3}}{67822307284 9 ^{4} 4 67822307284 9 ^{3}} + \frac{x ^{5}}{67822307284 9 ^{5}}

= 7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + 237405 x - \frac{15504 x ^{\frac{5}{4}}}{4 282475249} + \frac{38760 x ^{\frac{3}{2}}}{678223072849} - \frac{77520 x 4 x ^{3}}{7 4 67822307284 9 ^{3}} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - \frac{7 ^{11} \cdot 167960 x ^{\frac{9}{4}}}{67822307284 9 ^{2} 4 678223072849} + \frac{1.89383 E 20 x ^{\frac{5}{2}}}{3628800 \cdot 67822307284 9 ^{\frac{5}{2}}} - \frac{6777791831720 x ^{2} 4 x ^{3}}{67822307284 9 ^{2} 4 67822307284 9 ^{3}} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - \frac{26589360 7 x ^{\frac{13}{4}}}{67822307284 9 ^{3}} + \frac{38760 x ^{\frac{7}{2}}}{67822307284 9 ^{3} \cdot 7} - \frac{260575728 x ^{3} 4 x ^{3}}{67822307284 9 ^{3} 4 67822307284 9 ^{3}} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - \frac{1140 x ^{\frac{17}{4}}}{67822307284 9 ^{4} \cdot 7 ^{\frac{1}{2}}} + \frac{190 x ^{\frac{9}{2}}}{67822307284 9 ^{4} 282475249} - \frac{140 x ^{4} 4 x ^{3}}{67822307284 9 ^{4} 4 67822307284 9 ^{3}} + \frac{x ^{5}}{67822307284 9 ^{5}}

7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + 237405 x - \frac{15504 x ^{\frac{5}{4}}}{4 282475249} + \frac{38760 x ^{\frac{3}{2}}}{678223072849} - \frac{77520 x 4 x ^{3}}{7 4 67822307284 9 ^{3}} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - \frac{7 ^{11} \cdot 167960 x ^{\frac{9}{4}}}{67822307284 9 ^{2} 4 678223072849} + \frac{1.89383 E 20 x ^{\frac{5}{2}}}{3628800 \cdot 67822307284 9 ^{\frac{5}{2}}} - \frac{6777791831720 x ^{2} 4 x ^{3}}{67822307284 9 ^{2} 4 67822307284 9 ^{3}} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - \frac{26589360 7 x ^{\frac{13}{4}}}{67822307284 9 ^{3}} + \frac{38760 x ^{\frac{7}{2}}}{67822307284 9 ^{3} \cdot 7} - \frac{260575728 x ^{3} 4 x ^{3}}{67822307284 9 ^{3} 4 67822307284 9 ^{3}} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - \frac{1140 x ^{\frac{17}{4}}}{67822307284 9 ^{4} \cdot 7 ^{\frac{1}{2}}} + \frac{190 x ^{\frac{9}{2}}}{67822307284 9 ^{4} 282475249} - \frac{140 x ^{4} 4 x ^{3}}{67822307284 9 ^{4} 4 67822307284 9 ^{3}} + \frac{x ^{5}}{67822307284 9 ^{5}}

بسّط:

7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + 237405 x - \frac{15504 4 x ^{5}}{4 282475249} + \frac{38760 x ^{3}}{678223072849} - \frac{77520 x 4 x ^{3}}{7 4 67822307284 9 ^{3}} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - \frac{7 ^{11} \cdot 167960 4 x ^{9}}{67822307284 9 ^{2} 4 678223072849} + \frac{1.89383 E 20 x ^{5}}{3628800 67822307284 9 ^{5}} - \frac{6777791831720 x ^{2} 4 x ^{3}}{67822307284 9 ^{2} 4 67822307284 9 ^{3}} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - \frac{26589360 7 4 x ^{13}}{67822307284 9 ^{3}} + \frac{38760 x ^{7}}{67822307284 9 ^{3} \cdot 7} - \frac{260575728 x ^{3} 4 x ^{3}}{67822307284 9 ^{3} 4 67822307284 9 ^{3}} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - \frac{1140 4 x ^{17}}{67822307284 9 ^{4} 7} + \frac{190 x ^{9}}{67822307284 9 ^{4} 282475249} - \frac{140 x ^{4} 4 x ^{3}}{67822307284 9 ^{4} 4 67822307284 9 ^{3}} + \frac{x ^{5}}{67822307284 9 ^{5}}

= 7^{20} - \frac{7 ^{19} \cdot 20 4 x}{4 678223072849} + 7^{18} \cdot 190 (4 \frac{x}{678223072849})^{2} - 7^{17} \cdot 1140 (4 \frac{x}{678223072849})^{3} + 237405 x - \frac{15504 4 x ^{5}}{4 282475249} + \frac{38760 x ^{3}}{678223072849} - \frac{77520 x 4 x ^{3}}{7 4 67822307284 9 ^{3}} + \frac{7 ^{12} \cdot 125970 x ^{2}}{67822307284 9 ^{2}} - \frac{7 ^{11} \cdot 167960 4 x ^{9}}{67822307284 9 ^{2} 4 678223072849} + \frac{1.89383 E 20 x ^{5}}{3628800 67822307284 9 ^{5}} - \frac{6777791831720 x ^{2} 4 x ^{3}}{67822307284 9 ^{2} 4 67822307284 9 ^{3}} + \frac{726191981970 x ^{3}}{67822307284 9 ^{3}} - \frac{26589360 7 4 x ^{13}}{67822307284 9 ^{3}} + \frac{38760 x ^{7}}{67822307284 9 ^{3} \cdot 7} - \frac{260575728 x ^{3} 4 x ^{3}}{67822307284 9 ^{3} 4 67822307284 9 ^{3}} + \frac{11632845 x ^{4}}{67822307284 9 ^{4}} - \frac{1140 4 x ^{17}}{67822307284 9 ^{4} 7} + \frac{190 x ^{9}}{67822307284 9 ^{4} 282475249} - \frac{140 x ^{4} 4 x ^{3}}{67822307284 9 ^{4} 4 67822307284 9 ^{3}} + \frac{x ^{5}}{67822307284 9 ^{5}}

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