Determine if each of the following is always rational or always irrational. Justify your answers.
- The sum of two rational numbers.
- The sum of two irrational numbers.
- The sum of a rational and an irrational number.
Is it possible to have two rational numbers, \(a\) and \(b\), such that \(a^b\) is irrational? If so, display an example of such \(a\) and \(b\). If not, prove that it is not possible.
Decide if it is possible to have two irrational numbers, \(a\) and \(b\), such that \(a^b\) is rational. Prove it in either case.
Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)