I was thinking about the classic question, "Which is bigger, $e^\pi$ or $\pi^e$?" (no calculator), and I tried to create a question that is similar, but resistant to the usual methods used for the classic question. Here is what I came up with:
Which is bigger, ${\sqrt2}^9$ or $9^{\sqrt2}$ ?
The methods used for the classic question take advantage of the special properties of $e$ (for example, the maximum value of $y=\frac{\log x}{x}$ occurs at $x=e$). But in my question there is no $e$, so it seems that some other method must be used.
I can imagine a solution using Maclaurin series, with a lot of tedious pen-and-paper calculation. But I am mostly interested in solutions that require a minimum of calculation.
(According to my calculcator, $\sqrt2^9\approx 22.63$ and $9^\sqrt2\approx 22.36$.)