What is expand (1-e^{-x})(1-e^{-y}) ?

The solution to expand (1-e^{-x})(1-e^{-y}) is 1-e^{-y}-e^{-x}+e^{-x-y}

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expand (1-e^{-x})(1-e^{-y})

expand

(1 - e^{- x}) (1 - e^{- y})

1 - e^{- y} - e^{- x} + e^{- x - y}

Solution steps

(1 - e^{- x}) (1 - e^{- y})

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = - e^{- x}, c = 1, d = - e^{- y}

= 1 \cdot 1 + 1 \cdot (- e^{- y}) + (- e^{- x}) \cdot 1 + (- e^{- x}) (- e^{- y})

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 1 \cdot 1 - 1 \cdot e^{- y} - 1 \cdot e^{- x} + e^{- x} e^{- y}

Simplify

1 \cdot 1 - 1 \cdot e^{- y} - 1 \cdot e^{- x} + e^{- x} e^{- y} : 1 - e^{- y} - e^{- x} + e^{- x - y}

= 1 - e^{- y} - e^{- x} + e^{- x - y}

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What is expand (1-e^{-x})(1-e^{-y}) ?
The solution to expand (1-e^{-x})(1-e^{-y}) is 1-e^{-y}-e^{-x}+e^{-x-y}