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

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The discussion centers on the irrationality of π^(e) and e^(π), with e^(π) confirmed as transcendental and thus irrational based on the Gelfond-Schneider theorem. While Wolfram does not provide information on π^(e), the ability to perform arithmetic operations like addition, subtraction, multiplication, and division on these expressions remains valid regardless of their approximations. The approximate values of e^(π) at 21.7 and π^(e) at 21.2 are noted, but these approximations do not determine their rationality. The distinction between approximate values and actual properties of the numbers is emphasized, highlighting that approximation does not affect the classification of a number as rational or irrational. Therefore, the discussion concludes that while e^(π) is irrational, the status of π^(e) remains uncertain.
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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?
 
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RTCNTC said:
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.
 
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.
 
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