What is expand (2x+1/(x^2))^{12} ?

The solution to expand (2x+1/(x^2))^{12} is

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expand (2x+1/(x^2))^{12}

expand

(2 x + \frac{1}{x ^{2}})^{12}

4096 x^{12} + 24576 x^{9} + 67584 x^{6} + 112640 x^{3} + 126720 + \frac{101376}{x ^{3}} + \frac{59136}{x ^{6}} + \frac{25344}{x ^{9}} + \frac{7920}{x ^{12}} + \frac{1760}{x ^{15}} + \frac{264}{x ^{18}} + \frac{24}{x ^{21}} + \frac{1}{x ^{24}}

Solution steps

(2 x + \frac{1}{x ^{2}})^{12}

Apply binomial theorem:

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

a = 2 x, b = \frac{1}{x ^{2}}

= i = 0 \sum 12 (i 12) (2 x)^{(12 - i)} (\frac{1}{x ^{2}})^{i}

Expand summation

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

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (2 x)^{12} (\frac{1}{x ^{2}})^{0} : 4096 x^{12}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (2 x)^{11} (\frac{1}{x ^{2}})^{1} : 24576 x^{9}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (2 x)^{10} (\frac{1}{x ^{2}})^{2} : 67584 x^{6}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (2 x)^{9} (\frac{1}{x ^{2}})^{3} : 112640 x^{3}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (2 x)^{8} (\frac{1}{x ^{2}})^{4} : 126720

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (2 x)^{7} (\frac{1}{x ^{2}})^{5} : \frac{101376}{x ^{3}}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (2 x)^{6} (\frac{1}{x ^{2}})^{6} : \frac{59136}{x ^{6}}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (2 x)^{5} (\frac{1}{x ^{2}})^{7} : \frac{25344}{x ^{9}}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (2 x)^{4} (\frac{1}{x ^{2}})^{8} : \frac{7920}{x ^{12}}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (2 x)^{3} (\frac{1}{x ^{2}})^{9} : \frac{1760}{x ^{15}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (2 x)^{2} (\frac{1}{x ^{2}})^{10} : \frac{264}{x ^{18}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (2 x)^{1} (\frac{1}{x ^{2}})^{11} : \frac{24}{x ^{21}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (2 x)^{0} (\frac{1}{x ^{2}})^{12} : \frac{1}{x ^{24}}

= 4096 x^{12} + 24576 x^{9} + 67584 x^{6} + 112640 x^{3} + 126720 + \frac{101376}{x ^{3}} + \frac{59136}{x ^{6}} + \frac{25344}{x ^{9}} + \frac{7920}{x ^{12}} + \frac{1760}{x ^{15}} + \frac{264}{x ^{18}} + \frac{24}{x ^{21}} + \frac{1}{x ^{24}}

(w + 1) d w + (4 - w) d x

\frac{186624 x ( - 36 x ^{2} + 1 )}{( 1 + 36 x ^{2} ) ^{4}}

2 - p \cdot sin (x) (sin (a) + cos (a))

a (1 - cos (θ))

\frac{( x ^{2} + 3 ) ^{\frac{2}{3}} ( 3 x + 2 ) ^{2}}{x + 1}