Given a real number $x>1$ and a natural number $n$, what can we say about the integer part of $x^n$ in terms of $x$ and $n$?
For simplicity, let us assume $x<2$. For the first few values of $n$, $\left\lfloor x^n\right\rfloor =1$. But, once it crosses $1$, it starts growing quite rapidly. Is it possible to express $\left\lfloor x^n\right\rfloor$ in terms of $x$ and $n$? Or at least provide some bounds?
I was trying to think of it recursively. For a given $x>1$, let $I_n = \left\lfloor x^n\right\rfloor$ and $F_n = \left\{ x^n\right\}$ so that $$I_{n+1} = \left\lfloor x(I_n+F_n)\right\rfloor$$ but this doesn't take me anywhere.
Any useful results will be appreciated. Some specific sequences are given in OEIS A002379, OEIS A064628 and OEIS A091946.
Please note that the question is to find $\lfloor x^n \rfloor$ given a $x$ and an $n$ without using the $\text{floor}$ operator
If $\lfloor x^n\rfloor<2$, then $x^n<2$, so $x<2^{\frac1n}$.
If $\lfloor x^n\rfloor<k$, then $x^n<k$, so $x<k^{\frac1n}$.
So for a given $k$, find $n$ such that
$k^{\frac1{n+1}}<x<k^{\frac1n}$