Let us consider the number $$\Large\pi^{\pi^\pi}=\pi^{\pi\cdot\pi}=\pi^{\pi^2}$$
As the bases are equal, the exponents must be equal, So $$\pi=2$$
You can take any $x$ instead of $\pi$.
What is wrong in this proof?
Answer
Lets write $a \uparrow b$ to mean $a^b$.
Then the following reasoning is correct: $$(\pi \uparrow \pi)\uparrow \pi = \pi \uparrow (\pi \cdot \pi) = \pi \uparrow (\pi \uparrow 2)$$
However, we cannot necessarily deduce that the RHS equals
$$(\pi \uparrow \pi) \uparrow 2$$
because exponentiation isn't associative. Indeed, Google calculator tells me that:
$\pi \uparrow (\pi \uparrow 2) \approx 80662.6659386$
$(\pi \uparrow \pi) \uparrow 2 \approx 1329.48908322$
so if the calculator is correct to even the first decimal place, then
$$(\pi \uparrow \pi) \uparrow 2 \neq \pi \uparrow (\pi \uparrow 2).$$
Moral of the story: if in doubt, find better notation!
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