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شائع الجبر >

expand (81m^4+1/(3m))^{25}

الحلّ

expand

(81 m^{4} + \frac{1}{3 m})^{25}

الحلّ

8 1^{25} m^{100} + 3^{95} \cdot 25 m^{95} + 3^{91} \cdot 100 m^{90} + 3^{85} \cdot 2300 m^{85} + 3^{80} \cdot 12650 m^{80} + 3^{76} \cdot 17710 m^{75} + 3^{70} \cdot 177100 m^{70} + 3^{65} \cdot 480700 m^{65} + 3^{60} \cdot 1081575 m^{60} + 3^{55} \cdot 2042975 m^{55} + 3^{50} \cdot 3268760 m^{50} + 3^{45} \cdot 4457400 m^{45} + 3^{40} \cdot 5200300 m^{40} + 3^{35} \cdot 5200300 m^{35} + 3^{30} \cdot 4457400 m^{30} + 3^{25} \cdot 3268760 m^{25} + 3^{20} \cdot 2042975 m^{20} + 1.55194 E 13 m^{15} + 28384854300 m^{10} + 43035300 m^{5} + 53130 + \frac{12650}{243 m ^{5}} + \frac{2300}{59049 m ^{10}} + \frac{100}{4782969 m ^{15}} + \frac{25}{3 ^{20} m ^{20}} + \frac{1}{3 ^{25} m ^{25}}

خطوات الحلّ

(81 m^{4} + \frac{1}{3 m})^{25}

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

:فعّل نظريّة ذات الحدّين لنيوتن

a = 81 m^{4}, b = \frac{1}{3 m}

= i = 0 \sum 25 (i 25) (81 m^{4})^{(25 - i)} (\frac{1}{3 m})^{i}

وسّع حاصل الجمع

= \frac{25 !}{0 ! ( 25 - 0 ) !} (81 m^{4})^{25} (\frac{1}{3 m})^{0} + \frac{25 !}{1 ! ( 25 - 1 ) !} (81 m^{4})^{24} (\frac{1}{3 m})^{1} + \frac{25 !}{2 ! ( 25 - 2 ) !} (81 m^{4})^{23} (\frac{1}{3 m})^{2} + \frac{25 !}{3 ! ( 25 - 3 ) !} (81 m^{4})^{22} (\frac{1}{3 m})^{3} + \frac{25 !}{4 ! ( 25 - 4 ) !} (81 m^{4})^{21} (\frac{1}{3 m})^{4} + \frac{25 !}{5 ! ( 25 - 5 ) !} (81 m^{4})^{20} (\frac{1}{3 m})^{5} + \frac{25 !}{6 ! ( 25 - 6 ) !} (81 m^{4})^{19} (\frac{1}{3 m})^{6} + \frac{25 !}{7 ! ( 25 - 7 ) !} (81 m^{4})^{18} (\frac{1}{3 m})^{7} + \frac{25 !}{8 ! ( 25 - 8 ) !} (81 m^{4})^{17} (\frac{1}{3 m})^{8} + \frac{25 !}{9 ! ( 25 - 9 ) !} (81 m^{4})^{16} (\frac{1}{3 m})^{9} + \frac{25 !}{10 ! ( 25 - 10 ) !} (81 m^{4})^{15} (\frac{1}{3 m})^{10} + \frac{25 !}{11 ! ( 25 - 11 ) !} (81 m^{4})^{14} (\frac{1}{3 m})^{11} + \frac{25 !}{12 ! ( 25 - 12 ) !} (81 m^{4})^{13} (\frac{1}{3 m})^{12} + \frac{25 !}{13 ! ( 25 - 13 ) !} (81 m^{4})^{12} (\frac{1}{3 m})^{13} + \frac{25 !}{14 ! ( 25 - 14 ) !} (81 m^{4})^{11} (\frac{1}{3 m})^{14} + \frac{25 !}{15 ! ( 25 - 15 ) !} (81 m^{4})^{10} (\frac{1}{3 m})^{15} + \frac{25 !}{16 ! ( 25 - 16 ) !} (81 m^{4})^{9} (\frac{1}{3 m})^{16} + \frac{25 !}{17 ! ( 25 - 17 ) !} (81 m^{4})^{8} (\frac{1}{3 m})^{17} + \frac{25 !}{18 ! ( 25 - 18 ) !} (81 m^{4})^{7} (\frac{1}{3 m})^{18} + \frac{25 !}{19 ! ( 25 - 19 ) !} (81 m^{4})^{6} (\frac{1}{3 m})^{19} + \frac{25 !}{20 ! ( 25 - 20 ) !} (81 m^{4})^{5} (\frac{1}{3 m})^{20} + \frac{25 !}{21 ! ( 25 - 21 ) !} (81 m^{4})^{4} (\frac{1}{3 m})^{21} + \frac{25 !}{22 ! ( 25 - 22 ) !} (81 m^{4})^{3} (\frac{1}{3 m})^{22} + \frac{25 !}{23 ! ( 25 - 23 ) !} (81 m^{4})^{2} (\frac{1}{3 m})^{23} + \frac{25 !}{24 ! ( 25 - 24 ) !} (81 m^{4})^{1} (\frac{1}{3 m})^{24} + \frac{25 !}{25 ! ( 25 - 25 ) !} (81 m^{4})^{0} (\frac{1}{3 m})^{25}

\frac{25 !}{0 ! ( 25 - 0 ) !} (81 m^{4})^{25} (\frac{1}{3 m})^{0}

بسّط:

8 1^{25} m^{100}

\frac{25 !}{1 ! ( 25 - 1 ) !} (81 m^{4})^{24} (\frac{1}{3 m})^{1}

بسّط:

3^{95} \cdot 25 m^{95}

\frac{25 !}{2 ! ( 25 - 2 ) !} (81 m^{4})^{23} (\frac{1}{3 m})^{2}

بسّط:

3^{91} \cdot 100 m^{90}

\frac{25 !}{3 ! ( 25 - 3 ) !} (81 m^{4})^{22} (\frac{1}{3 m})^{3}

بسّط:

3^{85} \cdot 2300 m^{85}

\frac{25 !}{4 ! ( 25 - 4 ) !} (81 m^{4})^{21} (\frac{1}{3 m})^{4}

بسّط:

3^{80} \cdot 12650 m^{80}

\frac{25 !}{5 ! ( 25 - 5 ) !} (81 m^{4})^{20} (\frac{1}{3 m})^{5}

بسّط:

3^{76} \cdot 17710 m^{75}

\frac{25 !}{6 ! ( 25 - 6 ) !} (81 m^{4})^{19} (\frac{1}{3 m})^{6}

بسّط:

3^{70} \cdot 177100 m^{70}

\frac{25 !}{7 ! ( 25 - 7 ) !} (81 m^{4})^{18} (\frac{1}{3 m})^{7}

بسّط:

3^{65} \cdot 480700 m^{65}

\frac{25 !}{8 ! ( 25 - 8 ) !} (81 m^{4})^{17} (\frac{1}{3 m})^{8}

بسّط:

3^{60} \cdot 1081575 m^{60}

\frac{25 !}{9 ! ( 25 - 9 ) !} (81 m^{4})^{16} (\frac{1}{3 m})^{9}

بسّط:

3^{55} \cdot 2042975 m^{55}

\frac{25 !}{10 ! ( 25 - 10 ) !} (81 m^{4})^{15} (\frac{1}{3 m})^{10}

بسّط:

3^{50} \cdot 3268760 m^{50}

\frac{25 !}{11 ! ( 25 - 11 ) !} (81 m^{4})^{14} (\frac{1}{3 m})^{11}

بسّط:

3^{45} \cdot 4457400 m^{45}

\frac{25 !}{12 ! ( 25 - 12 ) !} (81 m^{4})^{13} (\frac{1}{3 m})^{12}

بسّط:

3^{40} \cdot 5200300 m^{40}

\frac{25 !}{13 ! ( 25 - 13 ) !} (81 m^{4})^{12} (\frac{1}{3 m})^{13}

بسّط:

3^{35} \cdot 5200300 m^{35}

\frac{25 !}{14 ! ( 25 - 14 ) !} (81 m^{4})^{11} (\frac{1}{3 m})^{14}

بسّط:

3^{30} \cdot 4457400 m^{30}

\frac{25 !}{15 ! ( 25 - 15 ) !} (81 m^{4})^{10} (\frac{1}{3 m})^{15}

بسّط:

3^{25} \cdot 3268760 m^{25}

\frac{25 !}{16 ! ( 25 - 16 ) !} (81 m^{4})^{9} (\frac{1}{3 m})^{16}

بسّط:

3^{20} \cdot 2042975 m^{20}

\frac{25 !}{17 ! ( 25 - 17 ) !} (81 m^{4})^{8} (\frac{1}{3 m})^{17}

بسّط:

1.55194 E 13 m^{15}

\frac{25 !}{18 ! ( 25 - 18 ) !} (81 m^{4})^{7} (\frac{1}{3 m})^{18}

بسّط:

28384854300 m^{10}

\frac{25 !}{19 ! ( 25 - 19 ) !} (81 m^{4})^{6} (\frac{1}{3 m})^{19}

بسّط:

43035300 m^{5}

\frac{25 !}{20 ! ( 25 - 20 ) !} (81 m^{4})^{5} (\frac{1}{3 m})^{20}

بسّط:

53130

\frac{25 !}{21 ! ( 25 - 21 ) !} (81 m^{4})^{4} (\frac{1}{3 m})^{21}

بسّط:

\frac{12650}{243 m ^{5}}

\frac{25 !}{22 ! ( 25 - 22 ) !} (81 m^{4})^{3} (\frac{1}{3 m})^{22}

بسّط:

\frac{2300}{59049 m ^{10}}

\frac{25 !}{23 ! ( 25 - 23 ) !} (81 m^{4})^{2} (\frac{1}{3 m})^{23}

بسّط:

\frac{100}{4782969 m ^{15}}

\frac{25 !}{24 ! ( 25 - 24 ) !} (81 m^{4})^{1} (\frac{1}{3 m})^{24}

بسّط:

\frac{25}{3 ^{20} m ^{20}}

\frac{25 !}{25 ! ( 25 - 25 ) !} (81 m^{4})^{0} (\frac{1}{3 m})^{25}

بسّط:

\frac{1}{3 ^{25} m ^{25}}

= 8 1^{25} m^{100} + 3^{95} \cdot 25 m^{95} + 3^{91} \cdot 100 m^{90} + 3^{85} \cdot 2300 m^{85} + 3^{80} \cdot 12650 m^{80} + 3^{76} \cdot 17710 m^{75} + 3^{70} \cdot 177100 m^{70} + 3^{65} \cdot 480700 m^{65} + 3^{60} \cdot 1081575 m^{60} + 3^{55} \cdot 2042975 m^{55} + 3^{50} \cdot 3268760 m^{50} + 3^{45} \cdot 4457400 m^{45} + 3^{40} \cdot 5200300 m^{40} + 3^{35} \cdot 5200300 m^{35} + 3^{30} \cdot 4457400 m^{30} + 3^{25} \cdot 3268760 m^{25} + 3^{20} \cdot 2042975 m^{20} + 1.55194 E 13 m^{15} + 28384854300 m^{10} + 43035300 m^{5} + 53130 + \frac{12650}{243 m ^{5}} + \frac{2300}{59049 m ^{10}} + \frac{100}{4782969 m ^{15}} + \frac{25}{3 ^{20} m ^{20}} + \frac{1}{3 ^{25} m ^{25}}

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