Solve the equation which has X as base and as a power

Question

$X^2=16^X$ How should I start it?

Answer 1

Just an intuitive answer, but $16^x$ grows faster then $x^2$ on $[0, \infty)$ and as $16^0 > 0^2$ we get that $16^x>x^2$ on $[0, \infty)$. Hence we have no solutions on $[0, \infty)$.

Now notice that on $(-\infty,0)$, $16^x$ is increasing, while $x^2$ is decreasing, hence you can have at most $1$ root. Finally, since $x=-\frac{1}{2}$ is a root, it is the only solution