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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

expand (e^x+k)^5

expand

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

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

Solution steps

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

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 5 (i 5) (e^{x})^{(5 - i)} k^{i}

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

\frac{1}{2} (5 - \frac{1}{2} ln (5 - 2))

y d x + (y^{3} - x) d y

\frac{d ^{2}}{d y ^{2}} ((d) ((x - y) e^{x y}))

x e^{3 x} (D^{2} - 6 D + 9)

(4 x^{3} y + 9) d x + (x^{4} - 5) d y

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