What is expand (1+t*e^{-t})^3 ?

The solution to expand (1+t*e^{-t})^3 is 1+3e^{-t}t+3e^{-2t}t^2+e^{-3t}t^3

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expand (1+t*e^{-t})^3

expand

(1 + t \cdot e^{- t})^{3}

1 + 3 e^{- t} t + 3 e^{- 2 t} t^{2} + e^{- 3 t} t^{3}

Solution steps

(1 + t e^{- t})^{3}

= (1 + e^{- t} t)^{3}

Apply Perfect Cube Formula:

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

a = 1, b = t e^{- t}

= 1^{3} + 3 \cdot 1^{2} t e^{- t} + 3 \cdot 1 \cdot (t e^{- t})^{2} + (t e^{- t})^{3}

Simplify

1^{3} + 3 \cdot 1^{2} t e^{- t} + 3 \cdot 1 \cdot (t e^{- t})^{2} + (t e^{- t})^{3} : 1 + 3 e^{- t} t + 3 e^{- 2 t} t^{2} + e^{- 3 t} t^{3}

= 1 + 3 e^{- t} t + 3 e^{- 2 t} t^{2} + e^{- 3 t} t^{3}

(x + i y)^{2} + 1

5 (22 - 2)

(4 + 5 i)^{7}

\frac{40}{( x + 8 ) ( x + 0.5 )}

(z^{2} - 1) e^{z}

What is expand (1+t*e^{-t})^3 ?
The solution to expand (1+t*e^{-t})^3 is 1+3e^{-t}t+3e^{-2t}t^2+e^{-3t}t^3