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Simplifier les expressions suivantes :
a.
$\mathrm e^{-3x} \times \mathrm e^x$;
Corrigé
$\mathrm e^{-3x} \times \mathrm e^x = \mathrm e^{-3x+x} = \boxed{\mathrm e^{-2x}} = \boxed{\dfrac 1 {\mathrm e^{2x}}}$;
b.
$\dfrac{\mathrm e^{-3x}}{\mathrm e^x}$;
Corrigé
$\dfrac{\mathrm e^{-3x}}{\mathrm e^x} = \mathrm e^{-3x-x} = \boxed{\mathrm e^{-4x}} = \boxed{\dfrac 1{\mathrm e^{4x}}}$;
c.
$\dfrac{\mathrm e^{2x} \times \mathrm e^{-5x}}{\mathrm e^{-2x}}$;
Corrigé
$\dfrac{\mathrm e^{2x} \times \mathrm e^{-5x}}{\mathrm e^{-2x}} = \mathrm e^{2x-5x+2x}= \boxed{\mathrm e^{-x}} = \boxed{\dfrac 1 {\mathrm e^x}}$;
d.
$\dfrac{\mathrm e^x}{\left(\mathrm e^{3x}\right)^2}$.
Corrigé
$\dfrac{\mathrm e^{x}}{\left(\mathrm e^{3x}\right)} = \dfrac{\mathrm e^{x}}{\mathrm e^{6x}} = \mathrm e^{x - 6x} = \boxed{\mathrm e^{-5x}} = \boxed{\dfrac{1}{\mathrm e^{5x}}}$.
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