We had the following set of exercises with a group of secondary school students (in Ressun lukio).
Assume that a and b are positive real numbers. Then assume one of the following:
(i) a and b are both rational
(ii) a and b are both irrational
(iii) a is rational and b is irrational.
(a) can a + b be rational/irrational?
(b) can ab be rational/irrational?
(c) can [tex]a^b[/tex] be rational/irrational? In case (iii) consider also [tex]b^a[/tex].
This makes 20 exercises total.