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شائع الجبر >

expand (2x^5+1/x)^{24}

الحلّ

expand

(2 x^{5} + \frac{1}{x})^{24}

الحلّ

16777216 x^{120} + 201326592 x^{114} + 1157627904 x^{108} + 4244635648 x^{102} + 11142168576 x^{96} + 22284337152 x^{90} + 35283533824 x^{84} + 45364543488 x^{78} + 48199827456 x^{72} + 42844291072 x^{66} + 32133218304 x^{60} + 20448411648 x^{54} + 11076222976 x^{48} + 5112102912 x^{42} + 2008326144 x^{36} + 669442048 x^{30} + 188280576 x^{24} + 44301312 x^{18} + 8614144 x^{12} + 1360128 x^{6} + 170016 + \frac{16192}{x ^{6}} + \frac{1104}{x ^{12}} + \frac{48}{x ^{18}} + \frac{1}{x ^{24}}

خطوات الحلّ

(2 x^{5} + \frac{1}{x})^{24}

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

:فعّل نظريّة ذات الحدّين لنيوتن

a = 2 x^{5}, b = \frac{1}{x}

= i = 0 \sum 24 (i 24) (2 x^{5})^{(24 - i)} (\frac{1}{x})^{i}

وسّع حاصل الجمع

= \frac{24 !}{0 ! ( 24 - 0 ) !} (2 x^{5})^{24} (\frac{1}{x})^{0} + \frac{24 !}{1 ! ( 24 - 1 ) !} (2 x^{5})^{23} (\frac{1}{x})^{1} + \frac{24 !}{2 ! ( 24 - 2 ) !} (2 x^{5})^{22} (\frac{1}{x})^{2} + \frac{24 !}{3 ! ( 24 - 3 ) !} (2 x^{5})^{21} (\frac{1}{x})^{3} + \frac{24 !}{4 ! ( 24 - 4 ) !} (2 x^{5})^{20} (\frac{1}{x})^{4} + \frac{24 !}{5 ! ( 24 - 5 ) !} (2 x^{5})^{19} (\frac{1}{x})^{5} + \frac{24 !}{6 ! ( 24 - 6 ) !} (2 x^{5})^{18} (\frac{1}{x})^{6} + \frac{24 !}{7 ! ( 24 - 7 ) !} (2 x^{5})^{17} (\frac{1}{x})^{7} + \frac{24 !}{8 ! ( 24 - 8 ) !} (2 x^{5})^{16} (\frac{1}{x})^{8} + \frac{24 !}{9 ! ( 24 - 9 ) !} (2 x^{5})^{15} (\frac{1}{x})^{9} + \frac{24 !}{10 ! ( 24 - 10 ) !} (2 x^{5})^{14} (\frac{1}{x})^{10} + \frac{24 !}{11 ! ( 24 - 11 ) !} (2 x^{5})^{13} (\frac{1}{x})^{11} + \frac{24 !}{12 ! ( 24 - 12 ) !} (2 x^{5})^{12} (\frac{1}{x})^{12} + \frac{24 !}{13 ! ( 24 - 13 ) !} (2 x^{5})^{11} (\frac{1}{x})^{13} + \frac{24 !}{14 ! ( 24 - 14 ) !} (2 x^{5})^{10} (\frac{1}{x})^{14} + \frac{24 !}{15 ! ( 24 - 15 ) !} (2 x^{5})^{9} (\frac{1}{x})^{15} + \frac{24 !}{16 ! ( 24 - 16 ) !} (2 x^{5})^{8} (\frac{1}{x})^{16} + \frac{24 !}{17 ! ( 24 - 17 ) !} (2 x^{5})^{7} (\frac{1}{x})^{17} + \frac{24 !}{18 ! ( 24 - 18 ) !} (2 x^{5})^{6} (\frac{1}{x})^{18} + \frac{24 !}{19 ! ( 24 - 19 ) !} (2 x^{5})^{5} (\frac{1}{x})^{19} + \frac{24 !}{20 ! ( 24 - 20 ) !} (2 x^{5})^{4} (\frac{1}{x})^{20} + \frac{24 !}{21 ! ( 24 - 21 ) !} (2 x^{5})^{3} (\frac{1}{x})^{21} + \frac{24 !}{22 ! ( 24 - 22 ) !} (2 x^{5})^{2} (\frac{1}{x})^{22} + \frac{24 !}{23 ! ( 24 - 23 ) !} (2 x^{5})^{1} (\frac{1}{x})^{23} + \frac{24 !}{24 ! ( 24 - 24 ) !} (2 x^{5})^{0} (\frac{1}{x})^{24}

\frac{24 !}{0 ! ( 24 - 0 ) !} (2 x^{5})^{24} (\frac{1}{x})^{0}

بسّط:

16777216 x^{120}

\frac{24 !}{1 ! ( 24 - 1 ) !} (2 x^{5})^{23} (\frac{1}{x})^{1}

بسّط:

201326592 x^{114}

\frac{24 !}{2 ! ( 24 - 2 ) !} (2 x^{5})^{22} (\frac{1}{x})^{2}

بسّط:

1157627904 x^{108}

\frac{24 !}{3 ! ( 24 - 3 ) !} (2 x^{5})^{21} (\frac{1}{x})^{3}

بسّط:

4244635648 x^{102}

\frac{24 !}{4 ! ( 24 - 4 ) !} (2 x^{5})^{20} (\frac{1}{x})^{4}

بسّط:

11142168576 x^{96}

\frac{24 !}{5 ! ( 24 - 5 ) !} (2 x^{5})^{19} (\frac{1}{x})^{5}

بسّط:

22284337152 x^{90}

\frac{24 !}{6 ! ( 24 - 6 ) !} (2 x^{5})^{18} (\frac{1}{x})^{6}

بسّط:

35283533824 x^{84}

\frac{24 !}{7 ! ( 24 - 7 ) !} (2 x^{5})^{17} (\frac{1}{x})^{7}

بسّط:

45364543488 x^{78}

\frac{24 !}{8 ! ( 24 - 8 ) !} (2 x^{5})^{16} (\frac{1}{x})^{8}

بسّط:

48199827456 x^{72}

\frac{24 !}{9 ! ( 24 - 9 ) !} (2 x^{5})^{15} (\frac{1}{x})^{9}

بسّط:

42844291072 x^{66}

\frac{24 !}{10 ! ( 24 - 10 ) !} (2 x^{5})^{14} (\frac{1}{x})^{10}

بسّط:

32133218304 x^{60}

\frac{24 !}{11 ! ( 24 - 11 ) !} (2 x^{5})^{13} (\frac{1}{x})^{11}

بسّط:

20448411648 x^{54}

\frac{24 !}{12 ! ( 24 - 12 ) !} (2 x^{5})^{12} (\frac{1}{x})^{12}

بسّط:

11076222976 x^{48}

\frac{24 !}{13 ! ( 24 - 13 ) !} (2 x^{5})^{11} (\frac{1}{x})^{13}

بسّط:

5112102912 x^{42}

\frac{24 !}{14 ! ( 24 - 14 ) !} (2 x^{5})^{10} (\frac{1}{x})^{14}

بسّط:

2008326144 x^{36}

\frac{24 !}{15 ! ( 24 - 15 ) !} (2 x^{5})^{9} (\frac{1}{x})^{15}

بسّط:

669442048 x^{30}

\frac{24 !}{16 ! ( 24 - 16 ) !} (2 x^{5})^{8} (\frac{1}{x})^{16}

بسّط:

188280576 x^{24}

\frac{24 !}{17 ! ( 24 - 17 ) !} (2 x^{5})^{7} (\frac{1}{x})^{17}

بسّط:

44301312 x^{18}

\frac{24 !}{18 ! ( 24 - 18 ) !} (2 x^{5})^{6} (\frac{1}{x})^{18}

بسّط:

8614144 x^{12}

\frac{24 !}{19 ! ( 24 - 19 ) !} (2 x^{5})^{5} (\frac{1}{x})^{19}

بسّط:

1360128 x^{6}

\frac{24 !}{20 ! ( 24 - 20 ) !} (2 x^{5})^{4} (\frac{1}{x})^{20}

بسّط:

170016

\frac{24 !}{21 ! ( 24 - 21 ) !} (2 x^{5})^{3} (\frac{1}{x})^{21}

بسّط:

\frac{16192}{x ^{6}}

\frac{24 !}{22 ! ( 24 - 22 ) !} (2 x^{5})^{2} (\frac{1}{x})^{22}

بسّط:

\frac{1104}{x ^{12}}

\frac{24 !}{23 ! ( 24 - 23 ) !} (2 x^{5})^{1} (\frac{1}{x})^{23}

بسّط:

\frac{48}{x ^{18}}

\frac{24 !}{24 ! ( 24 - 24 ) !} (2 x^{5})^{0} (\frac{1}{x})^{24}

بسّط:

\frac{1}{x ^{24}}

= 16777216 x^{120} + 201326592 x^{114} + 1157627904 x^{108} + 4244635648 x^{102} + 11142168576 x^{96} + 22284337152 x^{90} + 35283533824 x^{84} + 45364543488 x^{78} + 48199827456 x^{72} + 42844291072 x^{66} + 32133218304 x^{60} + 20448411648 x^{54} + 11076222976 x^{48} + 5112102912 x^{42} + 2008326144 x^{36} + 669442048 x^{30} + 188280576 x^{24} + 44301312 x^{18} + 8614144 x^{12} + 1360128 x^{6} + 170016 + \frac{16192}{x ^{6}} + \frac{1104}{x ^{12}} + \frac{48}{x ^{18}} + \frac{1}{x ^{24}}

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