14 questions linked to/from Fastest way to check if $x^y > y^x$?
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Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I calculate, without calculator or similar device, the values of $\pi^e$ and $e^\pi$ in order to compare them?
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How to determine without calculator which is bigger, $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ or $\left(\frac{1}{3}\right)^{\frac{1}{2}}$

How can you determine which one of these numbers is bigger (without calculating): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
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Collection of Non-Trick Questions That Require Work to Answer

This question is inspired by a comment on a recent popular question: Which area is larger, the blue area, or the white area? Warning: Spoiler Below! - Don't keep reading if you want to solve this ...
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Which is larger, $e^\pi$ or $\pi^e$? [duplicate]

I don't know how to approach this. I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was ...
Prove ( using calculus) that $20.17^{20.16}<20.16^{20.17}$. How do I do that?
A question about exponential functions: $a^b>b^a$ for $b>a>e$ [closed]
How can we prove that for $b>a>e$ ($e$ being the Euler’s number), $a^b$ is greater than $b^a$?