MHB Are π^(e) and e^(π) irrational with decimal approximations of 21.7 and 21.2?

[mathdad]

Can we say that π^(e) and e^(π) are irrational? If so, why?

Can we add, subtract, divide and multiply π^(e) and e^(π)
if π is approximately 3.1 and e is approximately 2.7?

 

[I like Serena]

Can we say that π^(e) and e^(π) are irrational? If so, why?

Can we add, subtract, divide and multiply π^(e) and e^(π)
if π is approximately 3.1 and e is approximately 2.7?

Let's see what wolfram says...

Wolfram doesn't say anything about pi^e, so for now that's unknown.
But it says that e^pi is transcendental (implying it's irrational). And indeed that follows from the Gelfond-Schneider theorem.

We can for sure add, subtract, divide, and multiply pi^e and e^pi - whether we approximate them with rational numbers or not.

 

[HallsofIvy]

Since $$\pi$$ is approximately 3.1 and e is approximately 2.7, [math]e^{/pi}[/math] is approximately 2.7^{3.1}= 21.7 and \pi^e is approximately 3.1^{2.7}= 21.2. But the word "approximately" is important there! An approximate value tells us nothing about whether a number is rational or irrational.