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Popular >

expand (2/x+y^2)^{12}

Solution

expand

(\frac{2}{x} + y^{2})^{12}

Solution

\frac{4096}{x ^{12}} + \frac{24576 y ^{2}}{x ^{11}} + \frac{67584 y ^{4}}{x ^{10}} + \frac{112640 y ^{6}}{x ^{9}} + \frac{126720 y ^{8}}{x ^{8}} + \frac{101376 y ^{10}}{x ^{7}} + \frac{59136 y ^{12}}{x ^{6}} + \frac{25344 y ^{14}}{x ^{5}} + \frac{7920 y ^{16}}{x ^{4}} + \frac{1760 y ^{18}}{x ^{3}} + \frac{264 y ^{20}}{x ^{2}} + \frac{24 y ^{22}}{x} + y^{24}

Solution steps

(\frac{2}{x} + y^{2})^{12}

Apply binomial theorem:

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

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

= i = 0 \sum 12 (i 12) (\frac{2}{x})^{(12 - i)} (y^{2})^{i}

Expand summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{2}{x})^{12} (y^{2})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{2}{x})^{11} (y^{2})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{2}{x})^{10} (y^{2})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{2}{x})^{9} (y^{2})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{2}{x})^{8} (y^{2})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{2}{x})^{7} (y^{2})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{2}{x})^{6} (y^{2})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{2}{x})^{5} (y^{2})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{2}{x})^{4} (y^{2})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{2}{x})^{3} (y^{2})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{2}{x})^{2} (y^{2})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{2}{x})^{1} (y^{2})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{2}{x})^{0} (y^{2})^{12}

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{2}{x})^{12} (y^{2})^{0} : \frac{4096}{x ^{12}}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{2}{x})^{11} (y^{2})^{1} : \frac{24576 y ^{2}}{x ^{11}}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{2}{x})^{10} (y^{2})^{2} : \frac{67584 y ^{4}}{x ^{10}}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{2}{x})^{9} (y^{2})^{3} : \frac{112640 y ^{6}}{x ^{9}}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{2}{x})^{8} (y^{2})^{4} : \frac{126720 y ^{8}}{x ^{8}}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{2}{x})^{7} (y^{2})^{5} : \frac{101376 y ^{10}}{x ^{7}}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{2}{x})^{6} (y^{2})^{6} : \frac{59136 y ^{12}}{x ^{6}}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{2}{x})^{5} (y^{2})^{7} : \frac{25344 y ^{14}}{x ^{5}}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{2}{x})^{4} (y^{2})^{8} : \frac{7920 y ^{16}}{x ^{4}}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{2}{x})^{3} (y^{2})^{9} : \frac{1760 y ^{18}}{x ^{3}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{2}{x})^{2} (y^{2})^{10} : \frac{264 y ^{20}}{x ^{2}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{2}{x})^{1} (y^{2})^{11} : \frac{24 y ^{22}}{x}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{2}{x})^{0} (y^{2})^{12} : y^{24}

= \frac{4096}{x ^{12}} + \frac{24576 y ^{2}}{x ^{11}} + \frac{67584 y ^{4}}{x ^{10}} + \frac{112640 y ^{6}}{x ^{9}} + \frac{126720 y ^{8}}{x ^{8}} + \frac{101376 y ^{10}}{x ^{7}} + \frac{59136 y ^{12}}{x ^{6}} + \frac{25344 y ^{14}}{x ^{5}} + \frac{7920 y ^{16}}{x ^{4}} + \frac{1760 y ^{18}}{x ^{3}} + \frac{264 y ^{20}}{x ^{2}} + \frac{24 y ^{22}}{x} + y^{24}

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