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شائع الجبر >

expand (6-\sqrt[4]{x/(13060694016)})^{19}

الحلّ

expand

(6 - 4 \frac{x}{13060694016})^{19}

الحلّ

6^{19} - \frac{6 ^{18} \cdot 19 4 x}{4 13060694016} + 6^{17} \cdot 171 (4 \frac{x}{13060694016})^{2} - 6^{16} \cdot 969 (4 \frac{x}{13060694016})^{3} + 139536 x - 69768 x 4 \frac{x}{13060694016} + \frac{2261 x x}{3888 6} - \frac{8398 x 4 x ^{3}}{4 1306069401 6 ^{3}} + \frac{2.74209 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{5585744406528 x ^{2} 4 \frac{x}{13060694016}}{1306069401 6 ^{2}} + \frac{3325608 6 x ^{2} x}{1306069401 6 ^{2}} - \frac{126948736512 x ^{2} 4 x ^{3}}{1306069401 6 ^{2} 4 1306069401 6 ^{3}} + \frac{14105415168 x ^{3}}{1306069401 6 ^{3}} - \frac{1265870592 x ^{3} 4 \frac{x}{13060694016}}{1306069401 6 ^{3}} + \frac{323 6 x ^{3} x}{1306069401 6 ^{3}} - \frac{5023296 x ^{3} 4 x ^{3}}{1306069401 6 ^{3} 4 1306069401 6 ^{3}} + \frac{209304 x ^{4}}{1306069401 6 ^{4}} - \frac{6156 x ^{4} 4 \frac{x}{13060694016}}{1306069401 6 ^{4}} + \frac{19 x ^{4} x}{1306069401 6 ^{4} 362797056} - \frac{x ^{4} ( 4 \frac{x}{13060694016} ) ^{3}}{1306069401 6 ^{4}}

خطوات الحلّ

(6 - 4 \frac{x}{13060694016})^{19}

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

:فعّل نظريّة ذات الحدّين لنيوتن

a = 6, b = - 4 \frac{x}{13060694016}

= i = 0 \sum 19 (i 19) \cdot 6^{(19 - i)} (- 4 \frac{x}{13060694016})^{i}

وسّع حاصل الجمع

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

\frac{19 !}{0 ! ( 19 - 0 ) !} \cdot 6^{19} (- 4 \frac{x}{13060694016})^{0}

بسّط:

6^{19}

\frac{19 !}{1 ! ( 19 - 1 ) !} \cdot 6^{18} (- 4 \frac{x}{13060694016})^{1}

بسّط:

- \frac{6 ^{18} \cdot 19 4 x}{4 13060694016}

\frac{19 !}{2 ! ( 19 - 2 ) !} \cdot 6^{17} (- 4 \frac{x}{13060694016})^{2}

بسّط:

6^{17} \cdot 171 (4 \frac{x}{13060694016})^{2}

\frac{19 !}{3 ! ( 19 - 3 ) !} \cdot 6^{16} (- 4 \frac{x}{13060694016})^{3}

بسّط:

- 6^{16} \cdot 969 (4 \frac{x}{13060694016})^{3}

\frac{19 !}{4 ! ( 19 - 4 ) !} \cdot 6^{15} (- 4 \frac{x}{13060694016})^{4}

بسّط:

139536 x

\frac{19 !}{5 ! ( 19 - 5 ) !} \cdot 6^{14} (- 4 \frac{x}{13060694016})^{5}

بسّط:

- 6^{14} \cdot 11628 (4 \frac{x}{13060694016})^{5}

\frac{19 !}{6 ! ( 19 - 6 ) !} \cdot 6^{13} (- 4 \frac{x}{13060694016})^{6}

بسّط:

6^{13} \cdot 27132 (4 \frac{x}{13060694016})^{6}

\frac{19 !}{7 ! ( 19 - 7 ) !} \cdot 6^{12} (- 4 \frac{x}{13060694016})^{7}

بسّط:

- 6^{12} \cdot 50388 (4 \frac{x}{13060694016})^{7}

\frac{19 !}{8 ! ( 19 - 8 ) !} \cdot 6^{11} (- 4 \frac{x}{13060694016})^{8}

بسّط:

\frac{2.74209 E 13 x ^{2}}{1306069401 6 ^{2}}

\frac{19 !}{9 ! ( 19 - 9 ) !} \cdot 6^{10} (- 4 \frac{x}{13060694016})^{9}

بسّط:

- 5585744406528 (4 \frac{x}{13060694016})^{9}

\frac{19 !}{10 ! ( 19 - 10 ) !} \cdot 6^{9} (- 4 \frac{x}{13060694016})^{10}

بسّط:

930957401088 (4 \frac{x}{13060694016})^{10}

\frac{19 !}{11 ! ( 19 - 11 ) !} \cdot 6^{8} (- 4 \frac{x}{13060694016})^{11}

بسّط:

- 126948736512 (4 \frac{x}{13060694016})^{11}

\frac{19 !}{12 ! ( 19 - 12 ) !} \cdot 6^{7} (- 4 \frac{x}{13060694016})^{12}

بسّط:

\frac{14105415168 x ^{3}}{1306069401 6 ^{3}}

\frac{19 !}{13 ! ( 19 - 13 ) !} \cdot 6^{6} (- 4 \frac{x}{13060694016})^{13}

بسّط:

- 1265870592 (4 \frac{x}{13060694016})^{13}

\frac{19 !}{14 ! ( 19 - 14 ) !} \cdot 6^{5} (- 4 \frac{x}{13060694016})^{14}

بسّط:

90419328 (4 \frac{x}{13060694016})^{14}

\frac{19 !}{15 ! ( 19 - 15 ) !} \cdot 6^{4} (- 4 \frac{x}{13060694016})^{15}

بسّط:

- 5023296 (4 \frac{x}{13060694016})^{15}

\frac{19 !}{16 ! ( 19 - 16 ) !} \cdot 6^{3} (- 4 \frac{x}{13060694016})^{16}

بسّط:

\frac{209304 x ^{4}}{1306069401 6 ^{4}}

\frac{19 !}{17 ! ( 19 - 17 ) !} \cdot 6^{2} (- 4 \frac{x}{13060694016})^{17}

بسّط:

- 6156 (4 \frac{x}{13060694016})^{17}

\frac{19 !}{18 ! ( 19 - 18 ) !} \cdot 6^{1} (- 4 \frac{x}{13060694016})^{18}

بسّط:

114 (4 \frac{x}{13060694016})^{18}

\frac{19 !}{19 ! ( 19 - 19 ) !} \cdot 6^{0} (- 4 \frac{x}{13060694016})^{19}

بسّط:

- (4 \frac{x}{13060694016})^{19}

= 6^{19} - \frac{6 ^{18} \cdot 19 4 x}{4 13060694016} + 6^{17} \cdot 171 (4 \frac{x}{13060694016})^{2} - 6^{16} \cdot 969 (4 \frac{x}{13060694016})^{3} + 139536 x - 6^{14} \cdot 11628 (4 \frac{x}{13060694016})^{5} + 6^{13} \cdot 27132 (4 \frac{x}{13060694016})^{6} - 6^{12} \cdot 50388 (4 \frac{x}{13060694016})^{7} + \frac{2.74209 E 13 x ^{2}}{1306069401 6 ^{2}} - 5585744406528 (4 \frac{x}{13060694016})^{9} + 930957401088 (4 \frac{x}{13060694016})^{10} - 126948736512 (4 \frac{x}{13060694016})^{11} + \frac{14105415168 x ^{3}}{1306069401 6 ^{3}} - 1265870592 (4 \frac{x}{13060694016})^{13} + 90419328 (4 \frac{x}{13060694016})^{14} - 5023296 (4 \frac{x}{13060694016})^{15} + \frac{209304 x ^{4}}{1306069401 6 ^{4}} - 6156 (4 \frac{x}{13060694016})^{17} + 114 (4 \frac{x}{13060694016})^{18} - (4 \frac{x}{13060694016})^{19}

6^{19} - \frac{6 ^{18} \cdot 19 4 x}{4 13060694016} + 6^{17} \cdot 171 (4 \frac{x}{13060694016})^{2} - 6^{16} \cdot 969 (4 \frac{x}{13060694016})^{3} + 139536 x - 6^{14} \cdot 11628 (4 \frac{x}{13060694016})^{5} + 6^{13} \cdot 27132 (4 \frac{x}{13060694016})^{6} - 6^{12} \cdot 50388 (4 \frac{x}{13060694016})^{7} + \frac{2.74209 E 13 x ^{2}}{1306069401 6 ^{2}} - 5585744406528 (4 \frac{x}{13060694016})^{9} + 930957401088 (4 \frac{x}{13060694016})^{10} - 126948736512 (4 \frac{x}{13060694016})^{11} + \frac{14105415168 x ^{3}}{1306069401 6 ^{3}} - 1265870592 (4 \frac{x}{13060694016})^{13} + 90419328 (4 \frac{x}{13060694016})^{14} - 5023296 (4 \frac{x}{13060694016})^{15} + \frac{209304 x ^{4}}{1306069401 6 ^{4}} - 6156 (4 \frac{x}{13060694016})^{17} + 114 (4 \frac{x}{13060694016})^{18} - (4 \frac{x}{13060694016})^{19}

بسّط:

6^{19} - \frac{6 ^{18} \cdot 19 4 x}{4 13060694016} + 6^{17} \cdot 171 (4 \frac{x}{13060694016})^{2} - 6^{16} \cdot 969 (4 \frac{x}{13060694016})^{3} + 139536 x - 69768 x 4 \frac{x}{13060694016} + \frac{2261 x x}{3888 6} - \frac{8398 x 4 x ^{3}}{4 1306069401 6 ^{3}} + \frac{2.74209 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{5585744406528 x ^{2} 4 \frac{x}{13060694016}}{1306069401 6 ^{2}} + \frac{3325608 6 x ^{2} x}{1306069401 6 ^{2}} - \frac{126948736512 x ^{2} 4 x ^{3}}{1306069401 6 ^{2} 4 1306069401 6 ^{3}} + \frac{14105415168 x ^{3}}{1306069401 6 ^{3}} - \frac{1265870592 x ^{3} 4 \frac{x}{13060694016}}{1306069401 6 ^{3}} + \frac{323 6 x ^{3} x}{1306069401 6 ^{3}} - \frac{5023296 x ^{3} 4 x ^{3}}{1306069401 6 ^{3} 4 1306069401 6 ^{3}} + \frac{209304 x ^{4}}{1306069401 6 ^{4}} - \frac{6156 x ^{4} 4 \frac{x}{13060694016}}{1306069401 6 ^{4}} + \frac{19 x ^{4} x}{1306069401 6 ^{4} 362797056} - \frac{x ^{4} ( 4 \frac{x}{13060694016} ) ^{3}}{1306069401 6 ^{4}}

= 6^{19} - \frac{6 ^{18} \cdot 19 4 x}{4 13060694016} + 6^{17} \cdot 171 (4 \frac{x}{13060694016})^{2} - 6^{16} \cdot 969 (4 \frac{x}{13060694016})^{3} + 139536 x - 69768 x 4 \frac{x}{13060694016} + \frac{2261 x x}{3888 6} - \frac{8398 x 4 x ^{3}}{4 1306069401 6 ^{3}} + \frac{2.74209 E 13 x ^{2}}{1306069401 6 ^{2}} - \frac{5585744406528 x ^{2} 4 \frac{x}{13060694016}}{1306069401 6 ^{2}} + \frac{3325608 6 x ^{2} x}{1306069401 6 ^{2}} - \frac{126948736512 x ^{2} 4 x ^{3}}{1306069401 6 ^{2} 4 1306069401 6 ^{3}} + \frac{14105415168 x ^{3}}{1306069401 6 ^{3}} - \frac{1265870592 x ^{3} 4 \frac{x}{13060694016}}{1306069401 6 ^{3}} + \frac{323 6 x ^{3} x}{1306069401 6 ^{3}} - \frac{5023296 x ^{3} 4 x ^{3}}{1306069401 6 ^{3} 4 1306069401 6 ^{3}} + \frac{209304 x ^{4}}{1306069401 6 ^{4}} - \frac{6156 x ^{4} 4 \frac{x}{13060694016}}{1306069401 6 ^{4}} + \frac{19 x ^{4} x}{1306069401 6 ^{4} 362797056} - \frac{x ^{4} ( 4 \frac{x}{13060694016} ) ^{3}}{1306069401 6 ^{4}}

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