Answer
$x=0$ and $x=\dfrac{4}{3}$
Work Step by Step
$4x^{3}e^{-3x}-3x^{4}e^{-3x}=0$
Take out common factor $x^{3}e^{-3x}$:
$x^{3}e^{-3x}(4-3x)=0$
Set all three factors equal to $0$ and solve each individual equation:
$x^{3}=0$
Take the cubic root of both sides:
$\sqrt[3]{x^{3}}=\sqrt[3]{0}$
$x=0$
Move on to the second equation:
$e^{-3x}=0$
No value of $x$ makes this second equation true. It has no solution. Move on to the third and final one:
$4-3x=0$
Solve for $x$:
$3x=4$
$x=\dfrac{4}{3}$
Our two solutions are:
$x=0$ and $x=\dfrac{4}{3}$