What is e^{-3t}((e^t-e^{-t}))/2 ?

The solution to e^{-3t}((e^t-e^{-t}))/2 is (e^{2t}-1)/(2e^{4t)}

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simplify e^{-3t}((e^t-e^{-t}))/2

simplify

e^{- 3 t} \frac{( e ^{t} - e ^{- t} )}{2}

\frac{e ^{2 t} - 1}{2 e ^{4 t}}

Solution steps

e^{- 3 t} \frac{( e ^{t} - e ^{- t} )}{2}

Apply exponent rule:

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

= \frac{1}{e ^{3 t}} \cdot \frac{e ^{t} - e ^{- t}}{2}

Multiply fractions:

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

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

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

= \frac{e ^{t} - e ^{- t}}{2 e ^{3 t}}

Factor

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

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

Cancel

\frac{e ^{- t} ( e ^{2 t} - 1 )}{e ^{3 t} \cdot 2} : \frac{e ^{2 t} - 1}{e ^{4 t} \cdot 2}

= \frac{e ^{2 t} - 1}{e ^{4 t} \cdot 2}

= \frac{e ^{2 t} - 1}{2 e ^{4 t}}

70 e^{- \frac{3}{250} 0}

(0.025 \cdot 4.54 \cdot 1 0^{- 4}) \cdot 3 \cdot 176.12 \cdot 1000 \cdot 43.8

(1.05)^{\frac{1}{2}} (1.03)^{\frac{1}{2}}

1.38 \cdot 1 0^{- 23} \cdot ln (36)

\frac{1}{2} \cdot \frac{20}{32.2} \cdot (20 \cdot 1)^{2}

What is e^{-3t}((e^t-e^{-t}))/2 ?
The solution to e^{-3t}((e^t-e^{-t}))/2 is (e^{2t}-1)/(2e^{4t)}