What is expand (1+x/2)^6 ?

The solution to expand (1+x/2)^6 is 1+3x+(15x^2)/4+(5x^3)/2+(15x^4)/(16)+(3x^5)/(16)+(x^6)/(64)

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expand (1+x/2)^6

expand

(1 + \frac{x}{2})^{6}

1 + 3 x + \frac{15 x ^{2}}{4} + \frac{5 x ^{3}}{2} + \frac{15 x ^{4}}{16} + \frac{3 x ^{5}}{16} + \frac{x ^{6}}{64}

Solution steps

(1 + \frac{x}{2})^{6}

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 6 (i 6) \cdot 1^{(6 - i)} (\frac{x}{2})^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} (\frac{x}{2})^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} (\frac{x}{2})^{1} : 3 x

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} (\frac{x}{2})^{2} : \frac{15 x ^{2}}{4}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} (\frac{x}{2})^{3} : \frac{5 x ^{3}}{2}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} (\frac{x}{2})^{4} : \frac{15 x ^{4}}{16}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} (\frac{x}{2})^{5} : \frac{3 x ^{5}}{16}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} (\frac{x}{2})^{6} : \frac{x ^{6}}{64}

= 1 + 3 x + \frac{15 x ^{2}}{4} + \frac{5 x ^{3}}{2} + \frac{15 x ^{4}}{16} + \frac{3 x ^{5}}{16} + \frac{x ^{6}}{64}

14 x^{2} + 7 x + 7

2 lo g_{10} (1)

\frac{cos ( x )}{1 - sin ^{2} ( x )}

x^{3} + 2 x^{2} + 2 x

\frac{a ^{2} + 3 a - ab - 3 b}{a ^{2} - ab + 5 a - 5 b}

What is expand (1+x/2)^6 ?
The solution to expand (1+x/2)^6 is 1+3x+(15x^2)/4+(5x^3)/2+(15x^4)/(16)+(3x^5)/(16)+(x^6)/(64)