What is expand (27x^5+1/(3x))^{12} ?

The solution to expand (27x^5+1/(3x))^{12} is

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expand (27x^5+1/(3x))^{12}

expand

(27 x^{5} + \frac{1}{3 x})^{12}

2 7^{12} x^{60} + 2 7^{11} \cdot 4 x^{54} + 3^{29} \cdot 22 x^{48} + 3^{24} \cdot 220 x^{42} + 3^{22} \cdot 55 x^{36} + 34093003032 x^{30} + 491051484 x^{24} + 5196312 x^{18} + 40095 x^{12} + 220 x^{6} + \frac{22}{27} + \frac{4}{2187 x ^{6}} + \frac{1}{531441 x ^{12}}

Solution steps

(27 x^{5} + \frac{1}{3 x})^{12}

Apply binomial theorem:

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

a = 27 x^{5}, b = \frac{1}{3 x}

= i = 0 \sum 12 (i 12) (27 x^{5})^{(12 - i)} (\frac{1}{3 x})^{i}

Expand summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} (27 x^{5})^{12} (\frac{1}{3 x})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (27 x^{5})^{11} (\frac{1}{3 x})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (27 x^{5})^{10} (\frac{1}{3 x})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (27 x^{5})^{9} (\frac{1}{3 x})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (27 x^{5})^{8} (\frac{1}{3 x})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (27 x^{5})^{7} (\frac{1}{3 x})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (27 x^{5})^{6} (\frac{1}{3 x})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (27 x^{5})^{5} (\frac{1}{3 x})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (27 x^{5})^{4} (\frac{1}{3 x})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (27 x^{5})^{3} (\frac{1}{3 x})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (27 x^{5})^{2} (\frac{1}{3 x})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (27 x^{5})^{1} (\frac{1}{3 x})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (27 x^{5})^{0} (\frac{1}{3 x})^{12}

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (27 x^{5})^{12} (\frac{1}{3 x})^{0} : 2 7^{12} x^{60}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (27 x^{5})^{11} (\frac{1}{3 x})^{1} : 2 7^{11} \cdot 4 x^{54}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (27 x^{5})^{10} (\frac{1}{3 x})^{2} : 3^{29} \cdot 22 x^{48}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (27 x^{5})^{9} (\frac{1}{3 x})^{3} : 3^{24} \cdot 220 x^{42}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (27 x^{5})^{8} (\frac{1}{3 x})^{4} : 3^{22} \cdot 55 x^{36}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (27 x^{5})^{7} (\frac{1}{3 x})^{5} : 34093003032 x^{30}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (27 x^{5})^{6} (\frac{1}{3 x})^{6} : 491051484 x^{24}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (27 x^{5})^{5} (\frac{1}{3 x})^{7} : 5196312 x^{18}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (27 x^{5})^{4} (\frac{1}{3 x})^{8} : 40095 x^{12}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (27 x^{5})^{3} (\frac{1}{3 x})^{9} : 220 x^{6}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (27 x^{5})^{2} (\frac{1}{3 x})^{10} : \frac{22}{27}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (27 x^{5})^{1} (\frac{1}{3 x})^{11} : \frac{4}{2187 x ^{6}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (27 x^{5})^{0} (\frac{1}{3 x})^{12} : \frac{1}{531441 x ^{12}}

= 2 7^{12} x^{60} + 2 7^{11} \cdot 4 x^{54} + 3^{29} \cdot 22 x^{48} + 3^{24} \cdot 220 x^{42} + 3^{22} \cdot 55 x^{36} + 34093003032 x^{30} + 491051484 x^{24} + 5196312 x^{18} + 40095 x^{12} + 220 x^{6} + \frac{22}{27} + \frac{4}{2187 x ^{6}} + \frac{1}{531441 x ^{12}}

(x e^{x} + e^{x})^{2}

(10 - 2)^{2}

x^{5} - 10 + 3 (2 x^{0.2}) 3 x - 1 + 6

(x - 1) (2 - x) (1 - x^{2})

1250 - 0.5 x^{2} + \frac{100 ( 2500 - x )}{0.02 x + 1}

What is expand (27x^5+1/(3x))^{12} ?
The solution to expand (27x^5+1/(3x))^{12} is