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Popular >

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

Solution

expand

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

Solution

1 + 6 x + \frac{33 x ^{2}}{2} + \frac{55 x ^{3}}{2} + \frac{495 x ^{4}}{16} + \frac{99 x ^{5}}{4} + \frac{231 x ^{6}}{16} + \frac{99 x ^{7}}{16} + \frac{495 x ^{8}}{256} + \frac{55 x ^{9}}{128} + \frac{33 x ^{10}}{512} + \frac{3 x ^{11}}{512} + \frac{x ^{12}}{4096}

Solution steps

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

Apply binomial theorem:

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

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

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

= 1 + 6 x + \frac{33 x ^{2}}{2} + \frac{55 x ^{3}}{2} + \frac{495 x ^{4}}{16} + \frac{99 x ^{5}}{4} + \frac{231 x ^{6}}{16} + \frac{99 x ^{7}}{16} + \frac{495 x ^{8}}{256} + \frac{55 x ^{9}}{128} + \frac{33 x ^{10}}{512} + \frac{3 x ^{11}}{512} + \frac{x ^{12}}{4096}

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Frequently Asked Questions (FAQ)

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