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

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

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

expand

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

x^{24} + 6 x^{21} + \frac{33 x ^{18}}{2} + \frac{55 x ^{15}}{2} + \frac{495 x ^{12}}{16} + \frac{99 x ^{9}}{4} + \frac{231 x ^{6}}{16} + \frac{99 x ^{3}}{16} + \frac{495}{256} + \frac{55}{128 x ^{3}} + \frac{33}{512 x ^{6}} + \frac{3}{512 x ^{9}} + \frac{1}{4096 x ^{12}}

Solution steps

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

Apply binomial theorem:

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

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

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

Expand summation

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

Simplify

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

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{2})^{11} (\frac{1}{2 x})^{1} : 6 x^{21}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{2})^{10} (\frac{1}{2 x})^{2} : \frac{33 x ^{18}}{2}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{2})^{9} (\frac{1}{2 x})^{3} : \frac{55 x ^{15}}{2}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (x^{2})^{8} (\frac{1}{2 x})^{4} : \frac{495 x ^{12}}{16}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{2})^{7} (\frac{1}{2 x})^{5} : \frac{99 x ^{9}}{4}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{2})^{6} (\frac{1}{2 x})^{6} : \frac{231 x ^{6}}{16}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{2})^{5} (\frac{1}{2 x})^{7} : \frac{99 x ^{3}}{16}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (x^{2})^{4} (\frac{1}{2 x})^{8} : \frac{495}{256}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{2})^{3} (\frac{1}{2 x})^{9} : \frac{55}{128 x ^{3}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{2})^{2} (\frac{1}{2 x})^{10} : \frac{33}{512 x ^{6}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{2})^{1} (\frac{1}{2 x})^{11} : \frac{3}{512 x ^{9}}

Simplify

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

= x^{24} + 6 x^{21} + \frac{33 x ^{18}}{2} + \frac{55 x ^{15}}{2} + \frac{495 x ^{12}}{16} + \frac{99 x ^{9}}{4} + \frac{231 x ^{6}}{16} + \frac{99 x ^{3}}{16} + \frac{495}{256} + \frac{55}{128 x ^{3}} + \frac{33}{512 x ^{6}} + \frac{3}{512 x ^{9}} + \frac{1}{4096 x ^{12}}

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