Show that
$\lim\limits_{n\to+\infty}\frac{2^{n^3}}{n!5^{n^2}-n^n}=+\infty$
without using l'Hôpitals rule or any special means.
My first attempts were to use inequalities to approximate $n!$ and $n^n$ but I am really unsure what to do with $2^{n^3}$ and $5^{n^3}$. Any specific hints on how to transform this fraction?