Answer
$x=0$, or $x=0.5$.
Work Step by Step
$4^{(x^2)}=(2^2)^{(x^2)}=2^{2x^2}$, hence the equation becomes: \begin{align*}2^{2x^2}=2^x.\end{align*}
The base is same on the 2 sides on the equation (and it is not $1$ or $-1$), hence they will be equal if the exponents are equal. Thus \begin{align*}2x^2=x\\2x^2-x=0\\x(2x-1)=0.\end{align*} Therefore by the zero property rule $x=0$, or $2x-1=0\\2x=1\\x=0.5$.
