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Popular >

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

Solution

expand

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

Solution

4096 + 8192 x + \frac{22528 x ^{2}}{3} + \frac{112640 x ^{3}}{27} + \frac{14080 x ^{4}}{9} + \frac{11264 x ^{5}}{27} + \frac{19712 x ^{6}}{243} + \frac{2816 x ^{7}}{243} + \frac{880 x ^{8}}{729} + \frac{1760 x ^{9}}{19683} + \frac{88 x ^{10}}{19683} + \frac{8 x ^{11}}{59049} + \frac{x ^{12}}{531441}

Solution steps

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

Apply binomial theorem:

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

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

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

Expand summation

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

Simplify

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

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} \cdot 2^{11} (\frac{x}{3})^{1} : 8192 x

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} \cdot 2^{10} (\frac{x}{3})^{2} : \frac{22528 x ^{2}}{3}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} \cdot 2^{9} (\frac{x}{3})^{3} : \frac{112640 x ^{3}}{27}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} \cdot 2^{8} (\frac{x}{3})^{4} : \frac{14080 x ^{4}}{9}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} \cdot 2^{7} (\frac{x}{3})^{5} : \frac{11264 x ^{5}}{27}

Simplify

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

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} \cdot 2^{5} (\frac{x}{3})^{7} : \frac{2816 x ^{7}}{243}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} \cdot 2^{4} (\frac{x}{3})^{8} : \frac{880 x ^{8}}{729}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} \cdot 2^{3} (\frac{x}{3})^{9} : \frac{1760 x ^{9}}{19683}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} \cdot 2^{2} (\frac{x}{3})^{10} : \frac{88 x ^{10}}{19683}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} \cdot 2^{1} (\frac{x}{3})^{11} : \frac{8 x ^{11}}{59049}

Simplify

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

= 4096 + 8192 x + \frac{22528 x ^{2}}{3} + \frac{112640 x ^{3}}{27} + \frac{14080 x ^{4}}{9} + \frac{11264 x ^{5}}{27} + \frac{19712 x ^{6}}{243} + \frac{2816 x ^{7}}{243} + \frac{880 x ^{8}}{729} + \frac{1760 x ^{9}}{19683} + \frac{88 x ^{10}}{19683} + \frac{8 x ^{11}}{59049} + \frac{x ^{12}}{531441}

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Frequently Asked Questions (FAQ)

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