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Popular >

expand (1+x/2)^{10}

Solution

expand

(1 + \frac{x}{2})^{10}

Solution

1 + 5 x + \frac{45 x ^{2}}{4} + 15 x^{3} + \frac{105 x ^{4}}{8} + \frac{63 x ^{5}}{8} + \frac{105 x ^{6}}{32} + \frac{15 x ^{7}}{16} + \frac{45 x ^{8}}{256} + \frac{5 x ^{9}}{256} + \frac{x ^{10}}{1024}

Solution steps

(1 + \frac{x}{2})^{10}

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 10 (i 10) \cdot 1^{(10 - i)} (\frac{x}{2})^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 1^{10} (\frac{x}{2})^{0} : 1

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 1^{9} (\frac{x}{2})^{1} : 5 x

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 1^{8} (\frac{x}{2})^{2} : \frac{45 x ^{2}}{4}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 1^{7} (\frac{x}{2})^{3} : 15 x^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 1^{6} (\frac{x}{2})^{4} : \frac{105 x ^{4}}{8}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 1^{5} (\frac{x}{2})^{5} : \frac{63 x ^{5}}{8}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 1^{4} (\frac{x}{2})^{6} : \frac{105 x ^{6}}{32}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 1^{3} (\frac{x}{2})^{7} : \frac{15 x ^{7}}{16}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 1^{2} (\frac{x}{2})^{8} : \frac{45 x ^{8}}{256}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 1^{1} (\frac{x}{2})^{9} : \frac{5 x ^{9}}{256}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 1^{0} (\frac{x}{2})^{10} : \frac{x ^{10}}{1024}

= 1 + 5 x + \frac{45 x ^{2}}{4} + 15 x^{3} + \frac{105 x ^{4}}{8} + \frac{63 x ^{5}}{8} + \frac{105 x ^{6}}{32} + \frac{15 x ^{7}}{16} + \frac{45 x ^{8}}{256} + \frac{5 x ^{9}}{256} + \frac{x ^{10}}{1024}

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Frequently Asked Questions (FAQ)

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