What is expand (1+2x)^5 ?

The solution to expand (1+2x)^5 is 1+10x+40x^2+80x^3+80x^4+32x^5

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expand (1+2x)^5

expand

(1 + 2 x)^{5}

1 + 10 x + 40 x^{2} + 80 x^{3} + 80 x^{4} + 32 x^{5}

Solution steps

(1 + 2 x)^{5}

Apply binomial theorem:

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

a = 1, b = 2 x

= i = 0 \sum 5 (i 5) \cdot 1^{(5 - i)} (2 x)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (2 x)^{0} : 1

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (2 x)^{1} : 10 x

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (2 x)^{2} : 40 x^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (2 x)^{3} : 80 x^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (2 x)^{4} : 80 x^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (2 x)^{5} : 32 x^{5}

= 1 + 10 x + 40 x^{2} + 80 x^{3} + 80 x^{4} + 32 x^{5}

\frac{x ^{2} + 2 x}{x ^{2}}

- 13 + 5

- 18 - 4

\frac{43}{6}

\frac{43}{8}

What is expand (1+2x)^5 ?
The solution to expand (1+2x)^5 is 1+10x+40x^2+80x^3+80x^4+32x^5