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شائع الجبر >

expand (a^2+a)^{31}

الحلّ

expand

(a^{2} + a)^{31}

الحلّ

a^{62} + 31 a^{61} + 465 a^{60} + 4495 a^{59} + 31465 a^{58} + 169911 a^{57} + 736281 a^{56} + 2629575 a^{55} + 7888725 a^{54} + 20160075 a^{53} + 44352165 a^{52} + 84672315 a^{51} + 141120525 a^{50} + 206253075 a^{49} + \frac{2.31182 E 19 a ^{48}}{87178291200} + \frac{3.93009 E 20 a ^{47}}{1307674368000} + \frac{3.93009 E 20 a ^{46}}{1307674368000} + \frac{2.31182 E 19 a ^{45}}{87178291200} + 206253075 a^{44} + 141120525 a^{43} + 84672315 a^{42} + 44352165 a^{41} + 20160075 a^{40} + 7888725 a^{39} + 2629575 a^{38} + 736281 a^{37} + 169911 a^{36} + 31465 a^{35} + 4495 a^{34} + 465 a^{33} + 31 a^{32} + a^{31}

خطوات الحلّ

(a^{2} + a)^{31}

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

:فعّل نظريّة ذات الحدّين لنيوتن

a = a^{2}, b = a

= i = 0 \sum 31 (i 31) (a^{2})^{(31 - i)} a^{i}

وسّع حاصل الجمع

= \frac{31 !}{0 ! ( 31 - 0 ) !} (a^{2})^{31} a^{0} + \frac{31 !}{1 ! ( 31 - 1 ) !} (a^{2})^{30} a^{1} + \frac{31 !}{2 ! ( 31 - 2 ) !} (a^{2})^{29} a^{2} + \frac{31 !}{3 ! ( 31 - 3 ) !} (a^{2})^{28} a^{3} + \frac{31 !}{4 ! ( 31 - 4 ) !} (a^{2})^{27} a^{4} + \frac{31 !}{5 ! ( 31 - 5 ) !} (a^{2})^{26} a^{5} + \frac{31 !}{6 ! ( 31 - 6 ) !} (a^{2})^{25} a^{6} + \frac{31 !}{7 ! ( 31 - 7 ) !} (a^{2})^{24} a^{7} + \frac{31 !}{8 ! ( 31 - 8 ) !} (a^{2})^{23} a^{8} + \frac{31 !}{9 ! ( 31 - 9 ) !} (a^{2})^{22} a^{9} + \frac{31 !}{10 ! ( 31 - 10 ) !} (a^{2})^{21} a^{10} + \frac{31 !}{11 ! ( 31 - 11 ) !} (a^{2})^{20} a^{11} + \frac{31 !}{12 ! ( 31 - 12 ) !} (a^{2})^{19} a^{12} + \frac{31 !}{13 ! ( 31 - 13 ) !} (a^{2})^{18} a^{13} + \frac{31 !}{14 ! ( 31 - 14 ) !} (a^{2})^{17} a^{14} + \frac{31 !}{15 ! ( 31 - 15 ) !} (a^{2})^{16} a^{15} + \frac{31 !}{16 ! ( 31 - 16 ) !} (a^{2})^{15} a^{16} + \frac{31 !}{17 ! ( 31 - 17 ) !} (a^{2})^{14} a^{17} + \frac{31 !}{18 ! ( 31 - 18 ) !} (a^{2})^{13} a^{18} + \frac{31 !}{19 ! ( 31 - 19 ) !} (a^{2})^{12} a^{19} + \frac{31 !}{20 ! ( 31 - 20 ) !} (a^{2})^{11} a^{20} + \frac{31 !}{21 ! ( 31 - 21 ) !} (a^{2})^{10} a^{21} + \frac{31 !}{22 ! ( 31 - 22 ) !} (a^{2})^{9} a^{22} + \frac{31 !}{23 ! ( 31 - 23 ) !} (a^{2})^{8} a^{23} + \frac{31 !}{24 ! ( 31 - 24 ) !} (a^{2})^{7} a^{24} + \frac{31 !}{25 ! ( 31 - 25 ) !} (a^{2})^{6} a^{25} + \frac{31 !}{26 ! ( 31 - 26 ) !} (a^{2})^{5} a^{26} + \frac{31 !}{27 ! ( 31 - 27 ) !} (a^{2})^{4} a^{27} + \frac{31 !}{28 ! ( 31 - 28 ) !} (a^{2})^{3} a^{28} + \frac{31 !}{29 ! ( 31 - 29 ) !} (a^{2})^{2} a^{29} + \frac{31 !}{30 ! ( 31 - 30 ) !} (a^{2})^{1} a^{30} + \frac{31 !}{31 ! ( 31 - 31 ) !} (a^{2})^{0} a^{31}

\frac{31 !}{0 ! ( 31 - 0 ) !} (a^{2})^{31} a^{0}

بسّط:

a^{62}

\frac{31 !}{1 ! ( 31 - 1 ) !} (a^{2})^{30} a^{1}

بسّط:

31 a^{61}

\frac{31 !}{2 ! ( 31 - 2 ) !} (a^{2})^{29} a^{2}

بسّط:

465 a^{60}

\frac{31 !}{3 ! ( 31 - 3 ) !} (a^{2})^{28} a^{3}

بسّط:

4495 a^{59}

\frac{31 !}{4 ! ( 31 - 4 ) !} (a^{2})^{27} a^{4}

بسّط:

31465 a^{58}

\frac{31 !}{5 ! ( 31 - 5 ) !} (a^{2})^{26} a^{5}

بسّط:

169911 a^{57}

\frac{31 !}{6 ! ( 31 - 6 ) !} (a^{2})^{25} a^{6}

بسّط:

736281 a^{56}

\frac{31 !}{7 ! ( 31 - 7 ) !} (a^{2})^{24} a^{7}

بسّط:

2629575 a^{55}

\frac{31 !}{8 ! ( 31 - 8 ) !} (a^{2})^{23} a^{8}

بسّط:

7888725 a^{54}

\frac{31 !}{9 ! ( 31 - 9 ) !} (a^{2})^{22} a^{9}

بسّط:

20160075 a^{53}

\frac{31 !}{10 ! ( 31 - 10 ) !} (a^{2})^{21} a^{10}

بسّط:

44352165 a^{52}

\frac{31 !}{11 ! ( 31 - 11 ) !} (a^{2})^{20} a^{11}

بسّط:

84672315 a^{51}

\frac{31 !}{12 ! ( 31 - 12 ) !} (a^{2})^{19} a^{12}

بسّط:

141120525 a^{50}

\frac{31 !}{13 ! ( 31 - 13 ) !} (a^{2})^{18} a^{13}

بسّط:

206253075 a^{49}

\frac{31 !}{14 ! ( 31 - 14 ) !} (a^{2})^{17} a^{14}

بسّط:

\frac{2.31182 E 19 a ^{48}}{87178291200}

\frac{31 !}{15 ! ( 31 - 15 ) !} (a^{2})^{16} a^{15}

بسّط:

\frac{3.93009 E 20 a ^{47}}{1307674368000}

\frac{31 !}{16 ! ( 31 - 16 ) !} (a^{2})^{15} a^{16}

بسّط:

\frac{3.93009 E 20 a ^{46}}{1307674368000}

\frac{31 !}{17 ! ( 31 - 17 ) !} (a^{2})^{14} a^{17}

بسّط:

\frac{2.31182 E 19 a ^{45}}{87178291200}

\frac{31 !}{18 ! ( 31 - 18 ) !} (a^{2})^{13} a^{18}

بسّط:

206253075 a^{44}

\frac{31 !}{19 ! ( 31 - 19 ) !} (a^{2})^{12} a^{19}

بسّط:

141120525 a^{43}

\frac{31 !}{20 ! ( 31 - 20 ) !} (a^{2})^{11} a^{20}

بسّط:

84672315 a^{42}

\frac{31 !}{21 ! ( 31 - 21 ) !} (a^{2})^{10} a^{21}

بسّط:

44352165 a^{41}

\frac{31 !}{22 ! ( 31 - 22 ) !} (a^{2})^{9} a^{22}

بسّط:

20160075 a^{40}

\frac{31 !}{23 ! ( 31 - 23 ) !} (a^{2})^{8} a^{23}

بسّط:

7888725 a^{39}

\frac{31 !}{24 ! ( 31 - 24 ) !} (a^{2})^{7} a^{24}

بسّط:

2629575 a^{38}

\frac{31 !}{25 ! ( 31 - 25 ) !} (a^{2})^{6} a^{25}

بسّط:

736281 a^{37}

\frac{31 !}{26 ! ( 31 - 26 ) !} (a^{2})^{5} a^{26}

بسّط:

169911 a^{36}

\frac{31 !}{27 ! ( 31 - 27 ) !} (a^{2})^{4} a^{27}

بسّط:

31465 a^{35}

\frac{31 !}{28 ! ( 31 - 28 ) !} (a^{2})^{3} a^{28}

بسّط:

4495 a^{34}

\frac{31 !}{29 ! ( 31 - 29 ) !} (a^{2})^{2} a^{29}

بسّط:

465 a^{33}

\frac{31 !}{30 ! ( 31 - 30 ) !} (a^{2})^{1} a^{30}

بسّط:

31 a^{32}

\frac{31 !}{31 ! ( 31 - 31 ) !} (a^{2})^{0} a^{31}

بسّط:

a^{31}

= a^{62} + 31 a^{61} + 465 a^{60} + 4495 a^{59} + 31465 a^{58} + 169911 a^{57} + 736281 a^{56} + 2629575 a^{55} + 7888725 a^{54} + 20160075 a^{53} + 44352165 a^{52} + 84672315 a^{51} + 141120525 a^{50} + 206253075 a^{49} + \frac{2.31182 E 19 a ^{48}}{87178291200} + \frac{3.93009 E 20 a ^{47}}{1307674368000} + \frac{3.93009 E 20 a ^{46}}{1307674368000} + \frac{2.31182 E 19 a ^{45}}{87178291200} + 206253075 a^{44} + 141120525 a^{43} + 84672315 a^{42} + 44352165 a^{41} + 20160075 a^{40} + 7888725 a^{39} + 2629575 a^{38} + 736281 a^{37} + 169911 a^{36} + 31465 a^{35} + 4495 a^{34} + 465 a^{33} + 31 a^{32} + a^{31}

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