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Popular Algebra >

expand (3p+8)^{25}

Solution

expand

(3 p + 8)^{25}

Solution

3^{25} p^{25} + 3^{24} \cdot 200 p^{24} + 3^{23} \cdot 19200 p^{23} + 3^{22} \cdot 1177600 p^{22} + 3^{21} \cdot 51814400 p^{21} + 3^{20} \cdot 1740963840 p^{20} + 3^{19} \cdot 46425702400 p^{19} + \frac{1.96842 E 24 p ^{18}}{5040} + \frac{9.4484 E 25 p ^{17}}{40320} + \frac{4.28328 E 27 p ^{16}}{362880} + \frac{7.13879 E 26 p ^{15}}{14175} + \frac{4.07931 E 27 p ^{14}}{22275} + \frac{2.66515 E 29 p ^{13}}{467775} + \frac{7.10707 E 29 p ^{12}}{467775} + \frac{2 ^{34} \cdot 3.15189 E 19 p ^{11}}{155925} + 8^{15} \cdot 193017009240 p^{10} + 8^{16} \cdot 40211876925 p^{9} + 8^{17} \cdot 7096213575 p^{8} + 8^{18} \cdot 1051290900 p^{7} + 8^{19} \cdot 129105900 p^{6} + 8^{20} \cdot 12910590 p^{5} + 8^{21} \cdot 1024650 p^{4} + 8^{22} \cdot 62100 p^{3} + 8^{23} \cdot 2700 p^{2} + 8^{24} \cdot 75 p + 8^{25}

Solution steps

(3 p + 8)^{25}

Apply binomial theorem:

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

a = 3 p, b = 8

= i = 0 \sum 25 (i 25) (3 p)^{(25 - i)} \cdot 8^{i}

Expand summation

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

Simplify

\frac{25 !}{0 ! ( 25 - 0 ) !} (3 p)^{25} \cdot 8^{0} : 3^{25} p^{25}

Simplify

\frac{25 !}{1 ! ( 25 - 1 ) !} (3 p)^{24} \cdot 8^{1} : 3^{24} \cdot 200 p^{24}

Simplify

\frac{25 !}{2 ! ( 25 - 2 ) !} (3 p)^{23} \cdot 8^{2} : 3^{23} \cdot 19200 p^{23}

Simplify

\frac{25 !}{3 ! ( 25 - 3 ) !} (3 p)^{22} \cdot 8^{3} : 3^{22} \cdot 1177600 p^{22}

Simplify

\frac{25 !}{4 ! ( 25 - 4 ) !} (3 p)^{21} \cdot 8^{4} : 3^{21} \cdot 51814400 p^{21}

Simplify

\frac{25 !}{5 ! ( 25 - 5 ) !} (3 p)^{20} \cdot 8^{5} : 3^{20} \cdot 1740963840 p^{20}

Simplify

\frac{25 !}{6 ! ( 25 - 6 ) !} (3 p)^{19} \cdot 8^{6} : 3^{19} \cdot 46425702400 p^{19}

Simplify

\frac{25 !}{7 ! ( 25 - 7 ) !} (3 p)^{18} \cdot 8^{7} : \frac{1.96842 E 24 p ^{18}}{5040}

Simplify

\frac{25 !}{8 ! ( 25 - 8 ) !} (3 p)^{17} \cdot 8^{8} : \frac{9.4484 E 25 p ^{17}}{40320}

Simplify

\frac{25 !}{9 ! ( 25 - 9 ) !} (3 p)^{16} \cdot 8^{9} : \frac{4.28328 E 27 p ^{16}}{362880}

Simplify

\frac{25 !}{10 ! ( 25 - 10 ) !} (3 p)^{15} \cdot 8^{10} : \frac{7.13879 E 26 p ^{15}}{14175}

Simplify

\frac{25 !}{11 ! ( 25 - 11 ) !} (3 p)^{14} \cdot 8^{11} : \frac{2.85552 E 28 p ^{14}}{155925}

Simplify

\frac{25 !}{12 ! ( 25 - 12 ) !} (3 p)^{13} \cdot 8^{12} : \frac{8 ^{12} \cdot 3.97138 E 21 p ^{13}}{479001600}

Simplify

\frac{25 !}{13 ! ( 25 - 13 ) !} (3 p)^{12} \cdot 8^{13} : \frac{8 ^{13} \cdot 1.32379 E 21 p ^{12}}{479001600}

Simplify

\frac{25 !}{14 ! ( 25 - 14 ) !} (3 p)^{11} \cdot 8^{14} : \frac{2 ^{34} \cdot 3.15189 E 19 p ^{11}}{155925}

Simplify

\frac{25 !}{15 ! ( 25 - 15 ) !} (3 p)^{10} \cdot 8^{15} : 8^{15} \cdot 193017009240 p^{10}

Simplify

\frac{25 !}{16 ! ( 25 - 16 ) !} (3 p)^{9} \cdot 8^{16} : 8^{16} \cdot 40211876925 p^{9}

Simplify

\frac{25 !}{17 ! ( 25 - 17 ) !} (3 p)^{8} \cdot 8^{17} : 8^{17} \cdot 7096213575 p^{8}

Simplify

\frac{25 !}{18 ! ( 25 - 18 ) !} (3 p)^{7} \cdot 8^{18} : 8^{18} \cdot 1051290900 p^{7}

Simplify

\frac{25 !}{19 ! ( 25 - 19 ) !} (3 p)^{6} \cdot 8^{19} : 8^{19} \cdot 129105900 p^{6}

Simplify

\frac{25 !}{20 ! ( 25 - 20 ) !} (3 p)^{5} \cdot 8^{20} : 8^{20} \cdot 12910590 p^{5}

Simplify

\frac{25 !}{21 ! ( 25 - 21 ) !} (3 p)^{4} \cdot 8^{21} : 8^{21} \cdot 1024650 p^{4}

Simplify

\frac{25 !}{22 ! ( 25 - 22 ) !} (3 p)^{3} \cdot 8^{22} : 8^{22} \cdot 62100 p^{3}

Simplify

\frac{25 !}{23 ! ( 25 - 23 ) !} (3 p)^{2} \cdot 8^{23} : 8^{23} \cdot 2700 p^{2}

Simplify

\frac{25 !}{24 ! ( 25 - 24 ) !} (3 p)^{1} \cdot 8^{24} : 8^{24} \cdot 75 p

Simplify

\frac{25 !}{25 ! ( 25 - 25 ) !} (3 p)^{0} \cdot 8^{25} : 8^{25}

= 3^{25} p^{25} + 3^{24} \cdot 200 p^{24} + 3^{23} \cdot 19200 p^{23} + 3^{22} \cdot 1177600 p^{22} + 3^{21} \cdot 51814400 p^{21} + 3^{20} \cdot 1740963840 p^{20} + 3^{19} \cdot 46425702400 p^{19} + \frac{1.96842 E 24 p ^{18}}{5040} + \frac{9.4484 E 25 p ^{17}}{40320} + \frac{4.28328 E 27 p ^{16}}{362880} + \frac{7.13879 E 26 p ^{15}}{14175} + \frac{2.85552 E 28 p ^{14}}{155925} + \frac{8 ^{12} \cdot 3.97138 E 21 p ^{13}}{479001600} + \frac{8 ^{13} \cdot 1.32379 E 21 p ^{12}}{479001600} + \frac{2 ^{34} \cdot 3.15189 E 19 p ^{11}}{155925} + 8^{15} \cdot 193017009240 p^{10} + 8^{16} \cdot 40211876925 p^{9} + 8^{17} \cdot 7096213575 p^{8} + 8^{18} \cdot 1051290900 p^{7} + 8^{19} \cdot 129105900 p^{6} + 8^{20} \cdot 12910590 p^{5} + 8^{21} \cdot 1024650 p^{4} + 8^{22} \cdot 62100 p^{3} + 8^{23} \cdot 2700 p^{2} + 8^{24} \cdot 75 p + 8^{25}

\frac{2.85552 E 28 p ^{14}}{155925} = \frac{4.07931 E 27 p ^{14}}{22275}

\frac{8 ^{12} \cdot 3.97138 E 21 p ^{13}}{479001600} = \frac{2.66515 E 29 p ^{13}}{467775}

\frac{8 ^{13} \cdot 1.32379 E 21 p ^{12}}{479001600} = \frac{7.10707 E 29 p ^{12}}{467775}

= 3^{25} p^{25} + 3^{24} \cdot 200 p^{24} + 3^{23} \cdot 19200 p^{23} + 3^{22} \cdot 1177600 p^{22} + 3^{21} \cdot 51814400 p^{21} + 3^{20} \cdot 1740963840 p^{20} + 3^{19} \cdot 46425702400 p^{19} + \frac{1.96842 E 24 p ^{18}}{5040} + \frac{9.4484 E 25 p ^{17}}{40320} + \frac{4.28328 E 27 p ^{16}}{362880} + \frac{7.13879 E 26 p ^{15}}{14175} + \frac{4.07931 E 27 p ^{14}}{22275} + \frac{2.66515 E 29 p ^{13}}{467775} + \frac{7.10707 E 29 p ^{12}}{467775} + \frac{2 ^{34} \cdot 3.15189 E 19 p ^{11}}{155925} + 8^{15} \cdot 193017009240 p^{10} + 8^{16} \cdot 40211876925 p^{9} + 8^{17} \cdot 7096213575 p^{8} + 8^{18} \cdot 1051290900 p^{7} + 8^{19} \cdot 129105900 p^{6} + 8^{20} \cdot 12910590 p^{5} + 8^{21} \cdot 1024650 p^{4} + 8^{22} \cdot 62100 p^{3} + 8^{23} \cdot 2700 p^{2} + 8^{24} \cdot 75 p + 8^{25}

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