For any integer $x$, let $f(x)=64x^6-27$. Now, write $f(x)=g(x)h(x)$, where $g(x)$ is the squarefree part of $f(x)$ and $h(x)$ is its squarefull (powerful) part. For example, $f(5)=13^2\cdot 61\cdot 97$ and so $g(5)=61\cdot 97$, while $h(5)=13^2$. Note that $h(5)^2>g(5)$.
I would like to find a $\theta\in (1,2)$, maybe $\theta=3/2$ (?) for which $h(x)^{\theta}<g(x)$, for all integer $x\geq 5$.
To prove that, I tried to use LTE, since $\nu_p(64x^6-27)=\nu_p(4x^2-3)$, for all $3<p\mid 4x^2-3$. Also, I tried to combine Zsigmondy theorem, but without success.
I also stress that $f(x)$ is squarefree for about 90% of positive integers $x\in [6,300]$ and so the desired bound holds since, in this case, $h(x)=1$.
Any suggestions?