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

The solution to expand (e^x+1)^5 is e^{5x}+5e^{4x}+10e^{3x}+10e^{2x}+5e^x+1

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expand (e^x+1)^5

expand

(e^{x} + 1)^{5}

e^{5 x} + 5 e^{4 x} + 10 e^{3 x} + 10 e^{2 x} + 5 e^{x} + 1

Solution steps

(e^{x} + 1)^{5}

Apply binomial theorem:

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

a = e^{x}, b = 1

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (e^{x})^{3} \cdot 1^{2} : 10 e^{3 x}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (e^{x})^{2} \cdot 1^{3} : 10 e^{2 x}

Simplify

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

\frac{5 !}{5 ! ( 5 - 5 ) !} (e^{x})^{0} \cdot 1^{5} = 1

= e^{5 x} + 5 e^{4 x} + 10 e^{3 x} + 10 e^{2 x} + 5 e^{x} + 1

(\frac{40 - 3 x}{4})^{2}

- \frac{2}{s ^{2} ( s - 2 )}

x \cdot (ln ∣ x ∣ + 1)

100000 (1 + sin (\frac{2 π ( t - 60 )}{365}))

\frac{d}{d t} ((q) 0.04 (700 - q))

What is expand (e^x+1)^5 ?
The solution to expand (e^x+1)^5 is e^{5x}+5e^{4x}+10e^{3x}+10e^{2x}+5e^x+1