What is 2e^{-t}(-e^{-2}+1) ?

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

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simplify 2e^{-t}(-e^{-2}+1)

simplify

2 e^{- t} (- e^{- 2} + 1)

\frac{2 ( e ^{2} - 1 )}{e ^{t + 2}}

Solution steps

2 e^{- t} (- e^{- 2} + 1)

Apply exponent rule:

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Apply exponent rule:

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

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

Join

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

= \frac{2 \cdot \frac{e ^{2} - 1}{e ^{2}}}{e ^{t}}

Multiply

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

= \frac{\frac{2 ( e ^{2} - 1 )}{e ^{2}}}{e ^{t}}

Apply the fraction rule:

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

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

e^{2} e^{t} = e^{2 + t}

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

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