Is there a formula or an algorithm to find the power of any integer that returns zero?

Question

Is there a formula or algorithm that can find the power ${^x}$ that returns $0$ for $n$ ${\mathbb N} $?

Answer 1

No such thing exists. Taking the example of $n = 2$, then $f(x) = 2^{-x}$ is a strictly decreasing function on $(0, \infty)$. You can see: $f(1) = \frac{1}{2}$, $f(2) = \frac{1}{4}$ etc etc, but only: $\lim\limits_{x \rightarrow \infty} f(x) = 0$.

Answer 2

No number, other than zero itself, has a solution to what you want. What I mean is that $n^x \neq 0$ unless $n = 0$. What you can do is just raising it to a very large negative power, e.g. with your example $2^{-13}\approx 0$, but $2^{-100}$ is even closer to zero, and $2^{-1000}$ is even closer etc. This is true for any $n > 0$.