What is e^{-x}(e^{-x}-xe^{-x}) ?

The solution to e^{-x}(e^{-x}-xe^{-x}) is (-x+1)/(e^{2x)}

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simplify e^{-x}(e^{-x}-xe^{-x})

simplify

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

\frac{- x + 1}{e ^{2 x}}

Solution steps

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

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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

= \frac{e ^{- x} - e ^{- x} x}{e ^{x}}

Factor out common term

e^{- x}

= \frac{e ^{- x} ( 1 - x )}{e ^{x}}

Cancel

\frac{e ^{- x} ( 1 - x )}{e ^{x}} : \frac{1 - x}{e ^{2 x}}

= \frac{1 - x}{e ^{2 x}}

= \frac{- x + 1}{e ^{2 x}}

\frac{( sin ( x ) ) ^{2}}{( cos ( x ) ) ^{2}}

(2)^{20}

0.3 \cdot 1 0^{- 3}

\frac{x + x}{x}

e^{2} e^{2}

What is e^{-x}(e^{-x}-xe^{-x}) ?
The solution to e^{-x}(e^{-x}-xe^{-x}) is (-x+1)/(e^{2x)}