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Popular Algebra >

expand (y^2+2/y)^{21}

Solution

expand

(y^{2} + \frac{2}{y})^{21}

Solution

y^{42} + 42 y^{39} + 840 y^{36} + 10640 y^{33} + 95760 y^{30} + 651168 y^{27} + 3472896 y^{24} + 14883840 y^{21} + 52093440 y^{18} + 150492160 y^{15} + 361181184 y^{12} + 722362368 y^{9} + 1203937280 y^{6} + 1666990080 y^{3} + 1905131520 + \frac{1778122752}{y ^{3}} + \frac{1333592064}{y ^{6}} + \frac{784465920}{y ^{9}} + \frac{348651520}{y ^{12}} + \frac{110100480}{y ^{15}} + \frac{22020096}{y ^{18}} + \frac{2097152}{y ^{21}}

Solution steps

(y^{2} + \frac{2}{y})^{21}

Apply binomial theorem:

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

a = y^{2}, b = \frac{2}{y}

= i = 0 \sum 21 (i 21) (y^{2})^{(21 - i)} (\frac{2}{y})^{i}

Expand summation

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

Simplify

\frac{21 !}{0 ! ( 21 - 0 ) !} (y^{2})^{21} (\frac{2}{y})^{0} : y^{42}

Simplify

\frac{21 !}{1 ! ( 21 - 1 ) !} (y^{2})^{20} (\frac{2}{y})^{1} : 42 y^{39}

Simplify

\frac{21 !}{2 ! ( 21 - 2 ) !} (y^{2})^{19} (\frac{2}{y})^{2} : 840 y^{36}

Simplify

\frac{21 !}{3 ! ( 21 - 3 ) !} (y^{2})^{18} (\frac{2}{y})^{3} : 10640 y^{33}

Simplify

\frac{21 !}{4 ! ( 21 - 4 ) !} (y^{2})^{17} (\frac{2}{y})^{4} : 95760 y^{30}

Simplify

\frac{21 !}{5 ! ( 21 - 5 ) !} (y^{2})^{16} (\frac{2}{y})^{5} : 651168 y^{27}

Simplify

\frac{21 !}{6 ! ( 21 - 6 ) !} (y^{2})^{15} (\frac{2}{y})^{6} : 3472896 y^{24}

Simplify

\frac{21 !}{7 ! ( 21 - 7 ) !} (y^{2})^{14} (\frac{2}{y})^{7} : 14883840 y^{21}

Simplify

\frac{21 !}{8 ! ( 21 - 8 ) !} (y^{2})^{13} (\frac{2}{y})^{8} : 52093440 y^{18}

Simplify

\frac{21 !}{9 ! ( 21 - 9 ) !} (y^{2})^{12} (\frac{2}{y})^{9} : 150492160 y^{15}

Simplify

\frac{21 !}{10 ! ( 21 - 10 ) !} (y^{2})^{11} (\frac{2}{y})^{10} : 361181184 y^{12}

Simplify

\frac{21 !}{11 ! ( 21 - 11 ) !} (y^{2})^{10} (\frac{2}{y})^{11} : 722362368 y^{9}

Simplify

\frac{21 !}{12 ! ( 21 - 12 ) !} (y^{2})^{9} (\frac{2}{y})^{12} : 1203937280 y^{6}

Simplify

\frac{21 !}{13 ! ( 21 - 13 ) !} (y^{2})^{8} (\frac{2}{y})^{13} : 1666990080 y^{3}

Simplify

\frac{21 !}{14 ! ( 21 - 14 ) !} (y^{2})^{7} (\frac{2}{y})^{14} : 1905131520

Simplify

\frac{21 !}{15 ! ( 21 - 15 ) !} (y^{2})^{6} (\frac{2}{y})^{15} : \frac{1778122752}{y ^{3}}

Simplify

\frac{21 !}{16 ! ( 21 - 16 ) !} (y^{2})^{5} (\frac{2}{y})^{16} : \frac{1333592064}{y ^{6}}

Simplify

\frac{21 !}{17 ! ( 21 - 17 ) !} (y^{2})^{4} (\frac{2}{y})^{17} : \frac{784465920}{y ^{9}}

Simplify

\frac{21 !}{18 ! ( 21 - 18 ) !} (y^{2})^{3} (\frac{2}{y})^{18} : \frac{348651520}{y ^{12}}

Simplify

\frac{21 !}{19 ! ( 21 - 19 ) !} (y^{2})^{2} (\frac{2}{y})^{19} : \frac{110100480}{y ^{15}}

Simplify

\frac{21 !}{20 ! ( 21 - 20 ) !} (y^{2})^{1} (\frac{2}{y})^{20} : \frac{22020096}{y ^{18}}

Simplify

\frac{21 !}{21 ! ( 21 - 21 ) !} (y^{2})^{0} (\frac{2}{y})^{21} : \frac{2097152}{y ^{21}}

= y^{42} + 42 y^{39} + 840 y^{36} + 10640 y^{33} + 95760 y^{30} + 651168 y^{27} + 3472896 y^{24} + 14883840 y^{21} + 52093440 y^{18} + 150492160 y^{15} + 361181184 y^{12} + 722362368 y^{9} + 1203937280 y^{6} + 1666990080 y^{3} + 1905131520 + \frac{1778122752}{y ^{3}} + \frac{1333592064}{y ^{6}} + \frac{784465920}{y ^{9}} + \frac{348651520}{y ^{12}} + \frac{110100480}{y ^{15}} + \frac{22020096}{y ^{18}} + \frac{2097152}{y ^{21}}

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