What is expand ((x^4)/2+4/(3x^2))^{12} ?

The solution to expand ((x^4)/2+4/(3x^2))^{12} is

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expand ((x^4)/2+4/(3x^2))^{12}

expand

(\frac{x ^{4}}{2} + \frac{4}{3 x ^{2}})^{12}

\frac{x ^{48}}{4096} + \frac{x ^{42}}{128} + \frac{11 x ^{36}}{96} + \frac{55 x ^{30}}{54} + \frac{55 x ^{24}}{9} + \frac{704 x ^{18}}{27} + \frac{19712 x ^{12}}{243} + \frac{45056 x ^{6}}{243} + \frac{225280}{729} + \frac{7208960}{19683 x ^{6}} + \frac{5767168}{19683 x ^{12}} + \frac{8388608}{59049 x ^{18}} + \frac{16777216}{531441 x ^{24}}

Solution steps

(\frac{x ^{4}}{2} + \frac{4}{3 x ^{2}})^{12}

Apply binomial theorem:

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

a = \frac{x ^{4}}{2}, b = \frac{4}{3 x ^{2}}

= i = 0 \sum 12 (i 12) (\frac{x ^{4}}{2})^{(12 - i)} (\frac{4}{3 x ^{2}})^{i}

Expand summation

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

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{x ^{4}}{2})^{12} (\frac{4}{3 x ^{2}})^{0} : \frac{x ^{48}}{4096}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{x ^{4}}{2})^{11} (\frac{4}{3 x ^{2}})^{1} : \frac{x ^{42}}{128}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{x ^{4}}{2})^{10} (\frac{4}{3 x ^{2}})^{2} : \frac{11 x ^{36}}{96}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{x ^{4}}{2})^{9} (\frac{4}{3 x ^{2}})^{3} : \frac{55 x ^{30}}{54}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{x ^{4}}{2})^{8} (\frac{4}{3 x ^{2}})^{4} : \frac{55 x ^{24}}{9}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{x ^{4}}{2})^{7} (\frac{4}{3 x ^{2}})^{5} : \frac{704 x ^{18}}{27}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{x ^{4}}{2})^{6} (\frac{4}{3 x ^{2}})^{6} : \frac{19712 x ^{12}}{243}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{x ^{4}}{2})^{5} (\frac{4}{3 x ^{2}})^{7} : \frac{45056 x ^{6}}{243}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{x ^{4}}{2})^{4} (\frac{4}{3 x ^{2}})^{8} : \frac{225280}{729}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{x ^{4}}{2})^{3} (\frac{4}{3 x ^{2}})^{9} : \frac{7208960}{19683 x ^{6}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{x ^{4}}{2})^{2} (\frac{4}{3 x ^{2}})^{10} : \frac{5767168}{19683 x ^{12}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{x ^{4}}{2})^{1} (\frac{4}{3 x ^{2}})^{11} : \frac{8388608}{59049 x ^{18}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{x ^{4}}{2})^{0} (\frac{4}{3 x ^{2}})^{12} : \frac{16777216}{531441 x ^{24}}

= \frac{x ^{48}}{4096} + \frac{x ^{42}}{128} + \frac{11 x ^{36}}{96} + \frac{55 x ^{30}}{54} + \frac{55 x ^{24}}{9} + \frac{704 x ^{18}}{27} + \frac{19712 x ^{12}}{243} + \frac{45056 x ^{6}}{243} + \frac{225280}{729} + \frac{7208960}{19683 x ^{6}} + \frac{5767168}{19683 x ^{12}} + \frac{8388608}{59049 x ^{18}} + \frac{16777216}{531441 x ^{24}}

(2 x + 3 y - 7) d x + (- 3 x - 2 y - 4) d y

\frac{1}{4 s t ( 1 + s ) ( 1 + t )}

100 ((0.75 \cdot 129218, 71) + (0, 25 \cdot 60301, 99))

(2 x + 3 + x^{2}) (2 x + 3 = x^{2})

4 y d x + (4 x + 4 x y cos (y)) d y

What is expand ((x^4)/2+4/(3x^2))^{12} ?
The solution to expand ((x^4)/2+4/(3x^2))^{12} is