Is there any way to represent irrational numbers with a finite amount of integers

Question

I know that rational numbers can be represented with two integers $\frac{a}{b}$.
But is there any way to represent irrational numbers with an finite amount of integers?

My best guess is $\frac{a}{b} ^ \frac{c}{d}$.
it can represent any root of any number, but I don't know if it can represent things like $\sqrt{2}^\sqrt{2}$.

Answer 1

There are uncountably many irrational numbers but there are only countably many finite sets (or lists) of integers. So it is impossible to represent every irrational number using only finitely many integers.