Is this a valid proof that $ e^{\pi^2} $ is irrational?

Question

Proving the irrationality of $e^n$

This essentially proves $e^n \ $ is irrational for all possible values of n and since $ e^{\pi^2} $ is essentially $e^{\pi\times\pi}$ and this fits the $e^n \ $ category so does that make it irrational?

Answer 1

Well, $e^{\pi^2}$ isn't just “essentially” $e^{\pi\times\pi}$; they're the same thing. And the question to which you have posted a link is about the irrationality of the numbers of the form $e^n$, with $n$ natural. Since $\pi^2\notin\mathbb N$, no, $e^{\pi^2}$ doesn't fit this category.

Answer 2

No. The answers to that question only show that $e^n$ is irrational for any positive integer $n$. This would only apply to $e^{\pi^2}$ if $\pi^2$ were a positive integer, which it is not.

Answer 3

No. The link shows that $e^n$ is irrational for any natural number $n$. But $\pi^2$ isn't a natural number, so that result doesn't apply. Note that there absolutely are exponents $x$ for which $e^x$ is rational - e.g. $e^{\ln 2}=2,$ and so on.