Answer
$x=1,2$
Work Step by Step
Re-write the given equation as: $(e)^{3x} \cdot (e)^{-2}=e^{x^2} ...(1)$
We know that $a^{m} \cdot a^n =a^{m+n}$
So, we can write equation (1) as: $e^{3x-2}=e^{x^2}$
Use the power rule: $a^p=a^q$.
We can see that the base $a=e$ is the same on both sides of the equation.
So, the exponents will also be equal.
This implies that $p=q$
Therefore, $3x-2=x^2 \\ x^2-3x+2=0 \\ (x-2)(x-1)=0 $
By the zero-product property, we have: $x=1,2$
