So I know that given some squarefree integer $n$, the fractional part of $\sqrt{n}$ can be bounded below by $\frac{1}{2\sqrt{n}+1}$ by
$$n=\left[\sqrt{n}\right]^2+2\left[\sqrt{n}\right]\left\{\sqrt{n}\right\}+\left\{\sqrt{n}\right\}^2$$
Where $[\sqrt{n}]$ and $\{\sqrt{n}\}$ denote the integer and fractional parts of $\sqrt{n}$.
My question is, can I find a stronger bound for square roots of integer powers? For example, it certainly seems that $\left\{n^{\frac{3}{2}}\right\}$ should have a sharper bound than $\frac{1}{2n^{\frac{3}{2}}}$, seeing as $n^3-1=m^2$ does not have integer solutions other than $n=1, m=0$.
Can I then extend this to other rational powers? For example, assuming $n$ does not contain a $d$-the power, can I say that $\left\{n^{\frac{s}{d}}\right\}$ is bounded by something nice?