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Popular >

expand (1+x/2)^{16}

Solution

expand

(1 + \frac{x}{2})^{16}

Solution

1 + 8 x + 30 x^{2} + 70 x^{3} + \frac{455 x ^{4}}{4} + \frac{273 x ^{5}}{2} + \frac{1001 x ^{6}}{8} + \frac{715 x ^{7}}{8} + \frac{6435 x ^{8}}{128} + \frac{715 x ^{9}}{32} + \frac{1001 x ^{10}}{128} + \frac{273 x ^{11}}{128} + \frac{455 x ^{12}}{1024} + \frac{35 x ^{13}}{512} + \frac{15 x ^{14}}{2048} + \frac{x ^{15}}{2048} + \frac{x ^{16}}{65536}

Solution steps

(1 + \frac{x}{2})^{16}

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 16 (i 16) \cdot 1^{(16 - i)} (\frac{x}{2})^{i}

Expand summation

= \frac{16 !}{0 ! ( 16 - 0 ) !} \cdot 1^{16} (\frac{x}{2})^{0} + \frac{16 !}{1 ! ( 16 - 1 ) !} \cdot 1^{15} (\frac{x}{2})^{1} + \frac{16 !}{2 ! ( 16 - 2 ) !} \cdot 1^{14} (\frac{x}{2})^{2} + \frac{16 !}{3 ! ( 16 - 3 ) !} \cdot 1^{13} (\frac{x}{2})^{3} + \frac{16 !}{4 ! ( 16 - 4 ) !} \cdot 1^{12} (\frac{x}{2})^{4} + \frac{16 !}{5 ! ( 16 - 5 ) !} \cdot 1^{11} (\frac{x}{2})^{5} + \frac{16 !}{6 ! ( 16 - 6 ) !} \cdot 1^{10} (\frac{x}{2})^{6} + \frac{16 !}{7 ! ( 16 - 7 ) !} \cdot 1^{9} (\frac{x}{2})^{7} + \frac{16 !}{8 ! ( 16 - 8 ) !} \cdot 1^{8} (\frac{x}{2})^{8} + \frac{16 !}{9 ! ( 16 - 9 ) !} \cdot 1^{7} (\frac{x}{2})^{9} + \frac{16 !}{10 ! ( 16 - 10 ) !} \cdot 1^{6} (\frac{x}{2})^{10} + \frac{16 !}{11 ! ( 16 - 11 ) !} \cdot 1^{5} (\frac{x}{2})^{11} + \frac{16 !}{12 ! ( 16 - 12 ) !} \cdot 1^{4} (\frac{x}{2})^{12} + \frac{16 !}{13 ! ( 16 - 13 ) !} \cdot 1^{3} (\frac{x}{2})^{13} + \frac{16 !}{14 ! ( 16 - 14 ) !} \cdot 1^{2} (\frac{x}{2})^{14} + \frac{16 !}{15 ! ( 16 - 15 ) !} \cdot 1^{1} (\frac{x}{2})^{15} + \frac{16 !}{16 ! ( 16 - 16 ) !} \cdot 1^{0} (\frac{x}{2})^{16}

Simplify

\frac{16 !}{0 ! ( 16 - 0 ) !} \cdot 1^{16} (\frac{x}{2})^{0} : 1

Simplify

\frac{16 !}{1 ! ( 16 - 1 ) !} \cdot 1^{15} (\frac{x}{2})^{1} : 8 x

Simplify

\frac{16 !}{2 ! ( 16 - 2 ) !} \cdot 1^{14} (\frac{x}{2})^{2} : 30 x^{2}

Simplify

\frac{16 !}{3 ! ( 16 - 3 ) !} \cdot 1^{13} (\frac{x}{2})^{3} : 70 x^{3}

Simplify

\frac{16 !}{4 ! ( 16 - 4 ) !} \cdot 1^{12} (\frac{x}{2})^{4} : \frac{455 x ^{4}}{4}

Simplify

\frac{16 !}{5 ! ( 16 - 5 ) !} \cdot 1^{11} (\frac{x}{2})^{5} : \frac{273 x ^{5}}{2}

Simplify

\frac{16 !}{6 ! ( 16 - 6 ) !} \cdot 1^{10} (\frac{x}{2})^{6} : \frac{1001 x ^{6}}{8}

Simplify

\frac{16 !}{7 ! ( 16 - 7 ) !} \cdot 1^{9} (\frac{x}{2})^{7} : \frac{715 x ^{7}}{8}

Simplify

\frac{16 !}{8 ! ( 16 - 8 ) !} \cdot 1^{8} (\frac{x}{2})^{8} : \frac{6435 x ^{8}}{128}

Simplify

\frac{16 !}{9 ! ( 16 - 9 ) !} \cdot 1^{7} (\frac{x}{2})^{9} : \frac{715 x ^{9}}{32}

Simplify

\frac{16 !}{10 ! ( 16 - 10 ) !} \cdot 1^{6} (\frac{x}{2})^{10} : \frac{1001 x ^{10}}{128}

Simplify

\frac{16 !}{11 ! ( 16 - 11 ) !} \cdot 1^{5} (\frac{x}{2})^{11} : \frac{273 x ^{11}}{128}

Simplify

\frac{16 !}{12 ! ( 16 - 12 ) !} \cdot 1^{4} (\frac{x}{2})^{12} : \frac{455 x ^{12}}{1024}

Simplify

\frac{16 !}{13 ! ( 16 - 13 ) !} \cdot 1^{3} (\frac{x}{2})^{13} : \frac{35 x ^{13}}{512}

Simplify

\frac{16 !}{14 ! ( 16 - 14 ) !} \cdot 1^{2} (\frac{x}{2})^{14} : \frac{15 x ^{14}}{2048}

Simplify

\frac{16 !}{15 ! ( 16 - 15 ) !} \cdot 1^{1} (\frac{x}{2})^{15} : \frac{x ^{15}}{2048}

Simplify

\frac{16 !}{16 ! ( 16 - 16 ) !} \cdot 1^{0} (\frac{x}{2})^{16} : \frac{x ^{16}}{65536}

= 1 + 8 x + 30 x^{2} + 70 x^{3} + \frac{455 x ^{4}}{4} + \frac{273 x ^{5}}{2} + \frac{1001 x ^{6}}{8} + \frac{715 x ^{7}}{8} + \frac{6435 x ^{8}}{128} + \frac{715 x ^{9}}{32} + \frac{1001 x ^{10}}{128} + \frac{273 x ^{11}}{128} + \frac{455 x ^{12}}{1024} + \frac{35 x ^{13}}{512} + \frac{15 x ^{14}}{2048} + \frac{x ^{15}}{2048} + \frac{x ^{16}}{65536}

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