The given conditions state that \( p \) and \( r \) are rational numbers and \( r \) is an irrational number. However, it seems there might be a misunderstanding since it is mentioned that \( r \) is both rational and irrational.
If we assume \( p \) is a rational number and \( r \) is an irrational number (not both), here are a few expressions that would represent an irrational number:
- \( p + r \) — The sum of a rational number and an irrational number is always irrational.
- \( p \cdot r \) — The product of a rational non-zero number and an irrational number is always irrational.
- \( r^2 \) — The square of an irrational number is also irrational.
- \( \sqrt{r} \) — If \( r \) is a positive irrational number, then \( \sqrt{r} \) is also irrational.
Among these examples, any of \( p + r \) or \( p \cdot r \) effectively illustrates that combining rational and irrational numbers results in an irrational number.