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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

expand (e^x+k)^4

expand

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

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

Solution steps

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

Apply binomial theorem:

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

a = e^{x}, b = k

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

1 + \frac{k}{j w ^{2} ( j w + 3.6 )}

\frac{x - 5}{( x + 3 ) ( x ^{2} + 7 x + 6 )}

\frac{2 x ( 216 - 216 x + 72 x ^{2} - 8 x ^{3} )}{81}

\frac{( x ^{2} + 3 x - 5 ) z ^{2} ( z - 3 ) ^{2} + 3}{x ^{2} + 3 x + 4}

\frac{e ^{t} ( t ^{2} + 1 ) - 2 t e ^{t}}{( t ^{2} + 1 ) ^{2}}

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