Why does 2 result in the same value regardless of whether it is added to itself, multiplied by itself, or put to the power of itself?

Question

I'm inferring that any hyperoperation you could apply here using two for every value would result in four. Why is this?

Answer 1

Suppose $n$ is an integer and $n+n=n\cdot n$.

Then $2n=n^2$ so $n^2-2n=0$. Factoring gives $n(n-2)=0$.

So either $n=0$ or $n=2$.

But if we also have that $n+n=n\cdot n=n^n$ then we have $n^2=n^n$ which gives $n=1$ or $n=2$, so $n=2$ is the common solution to the two equations.