What is expand (1+e^x)^4 ?

The solution to expand (1+e^x)^4 is 1+4e^x+6e^{2x}+4e^{3x}+e^{4x}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

expand (1+e^x)^4

expand

(1 + e^{x})^{4}

1 + 4 e^{x} + 6 e^{2 x} + 4 e^{3 x} + e^{4 x}

Solution steps

(1 + e^{x})^{4}

Apply binomial theorem:

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

a = 1, b = e^{x}

= i = 0 \sum 4 (i 4) \cdot 1^{(4 - i)} (e^{x})^{i}

Expand summation

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

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1^{4} (e^{x})^{0} = 1

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 1^{3} (e^{x})^{1} : 4 e^{x}

Simplify

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

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 1^{1} (e^{x})^{3} : 4 e^{3 x}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1^{0} (e^{x})^{4} : e^{4 x}

= 1 + 4 e^{x} + 6 e^{2 x} + 4 e^{3 x} + e^{4 x}

\frac{4 ( x + h ) - 5 - 4 x - 5}{h}

(a + b + c) (d + e)

3.42 e^{-} \cdot 79.7 (1 \cdot sin (225 x) + cos (225 x) \cdot 225 x)

\frac{\partial}{\partial x} ((u (x, t)) (3 x - 2 t))

\frac{3 x}{( 2 - x ) ( 4 + x ^{2} )}

What is expand (1+e^x)^4 ?
The solution to expand (1+e^x)^4 is 1+4e^x+6e^{2x}+4e^{3x}+e^{4x}