What is expand (e^{jθ}+e^{-jθ})^3 ?

The solution to expand (e^{jθ}+e^{-jθ})^3 is e^{3jθ}+3e^{jθ}+3e^{-jθ}+e^{-3jθ}

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expand (e^{jθ}+e^{-jθ})^3

expand

(e^{j θ} + e^{- j θ})^{3}

e^{3 j θ} + 3 e^{j θ} + 3 e^{- j θ} + e^{- 3 j θ}

Solution steps

(e^{j θ} + e^{- j θ})^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

a = e^{j θ}, b = e^{- j θ}

= (e^{j θ})^{3} + 3 (e^{j θ})^{2} e^{- j θ} + 3 e^{j θ} (e^{- j θ})^{2} + (e^{- j θ})^{3}

Simplify

(e^{j θ})^{3} + 3 (e^{j θ})^{2} e^{- j θ} + 3 e^{j θ} (e^{- j θ})^{2} + (e^{- j θ})^{3} : e^{3 j θ} + 3 e^{j θ} + 3 e^{- j θ} + e^{- 3 j θ}

= e^{3 j θ} + 3 e^{j θ} + 3 e^{- j θ} + e^{- 3 j θ}

(2 e^{- 4 t} - 2)^{3}

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

(x + Δ x) (x - 2)

\frac{s + 3}{s ( s ^{2} + 1 )}

(- 2 e^{- 4 t} - 2)^{3}

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