I'm a high school math student. My last math class was Algebra $2$ Honors, so I'm not super well-versed when it comes to set theory - that being said, since I first thought about this problem, it's been nagging at me.
Let there be two variables, a and b, defined like this:
$a \in \Bbb{R}$, $a \notin \Bbb{Z}$, and $b \in \Bbb{Z}$.
By those constraints, could the statement $a^b \in \Bbb{Z}$?
For the record, I know that $b^a$ can! For example, if $a = 1.5$ and $b = 9$, then $b^a = 9 \times 3 = 27$, and $27 \in \Bbb{Z}$.
EDIT: I saw an answer using $a = \root\of2$ and $b = 2$, which works perfectly! I do wonder, though: Could you extrapolate this any further?
For example, if $a \in \Bbb{Q}$, could there still be a set of values where $a^b \in \Bbb{Z}$? :)
Take $a=\sqrt{2}$ and $b=2$, $a^b=2\in Z$