# 1.E: Numbers - Real (ℝ) and Rational (ℚ) (Exercises)

- Page ID
- 7917

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### Q1

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.

### Q2

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.

### Q3

Decide if it is possible to have two irrational numbers, \(a\) and \(b\), such that \(a^b\) is rational. Prove it in either case.

### Contributor

Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)