What is expand (e^x-e^{-x})^5 ?

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

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expand (e^x-e^{-x})^5

expand

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

e^{5 x} - 5 e^{3 x} + 10 e^{x} - 10 e^{- x} + 5 e^{- 3 x} - e^{- 5 x}

Solution steps

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

Apply binomial theorem:

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

a = e^{x}, b = - e^{- x}

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

Expand summation

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

Simplify

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

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (e^{x})^{4} (- e^{- x})^{1} : - 5 e^{3 x}

Simplify

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

Simplify

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

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (e^{x})^{1} (- e^{- x})^{4} : 5 e^{- 3 x}

Simplify

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

= e^{5 x} - 5 e^{3 x} + 10 e^{x} - 10 e^{- x} + 5 e^{- 3 x} - e^{- 5 x}

\frac{( x ^{2} + 1 ) ^{3}}{6}

\frac{( x - 2 ) ( 2 x + 3 )}{x ^{2} - 1}

- \frac{5 x ( 4 x - 14 )}{3}

\frac{3 ( s + x )}{s ( s + 1 )}

\frac{10 ( 5 + 74 )}{49}

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