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Simplifier les expressions :
1.
$\exp(3x-1)\exp(7 - 4x)$;
Corrigé
\begin{align*}
\exp(3x-1)\cdot\exp(7-4x)
&= \exp(3x-1+7-4x)&
\\
&= \exp(6-x).&
\end{align*}
2.
$\dfrac 1 {\exp(5x)}$;
Corrigé
\[\dfrac 1 {\exp(5x)} = \exp(-5x)\].
3.
$\mathrm e^x\mathrm e^x$;
Corrigé
\[\mathrm e^x\mathrm e^x =
\mathrm e^{x+x}= \mathrm e^{2x}.\]
Variante
\[\mathrm e^x \mathrm e^x = \left(\mathrm e^x\right)^2
=\mathrm e^{2x}.\]
4.
$\dfrac{\mathrm e^{3x}}{\mathrm e^x}$;
Corrigé
\[\dfrac{\mathrm e^{3x}}{\mathrm e^x} = \mathrm e^{3x-x} = \mathrm e^{2x}.\]
5.
$4\exp(x)\exp(5x + 3)$;
Corrigé
\begin{align*}
4\exp(x)\exp(5x+3)
&= 4\exp(x+5x+3)&
\\
&= 4\exp(6x+3).&
\end{align*}
6.
$\dfrac{\exp(8x-4)}{\exp(4)\exp(x+2)}$;
Corrigé
\begin{align*}
\dfrac{\exp(8x-4)}{\exp(4)\exp(x+2)}
&= \exp(8x - 4 - 4 - x -2)&
\\
&=\exp(7x - 10).&
\end{align*}
7.
$\mathrm e^{-x}\left(\mathrm e^x\right)^4$;
Corrigé
\[\mathrm e^{-x}\left(\mathrm e^{x}\right)^4 = \mathrm e^{-x}\mathrm e^{4x}
=\mathrm e^{-x+4x} = \mathrm e^{3x}.\]
8.
$\dfrac{\mathrm e^{3x}\mathrm e^{2x}}{4\mathrm e^x}$.
Corrigé
\[\dfrac{\mathrm e^{3x}\mathrm e^{2x}}{4\mathrm e^x}=\dfrac{\mathrm e^{3x+2x-x}}{4}
=\dfrac{\mathrm e^{4x}}{4}.\]
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