To determine which expression represents a rational number given that \( p \) and \( q \) are rational numbers and \( r \) is an irrational number, we need to consider how the operations between these numbers affect their rationality.
Here are some possible combinations:
- \( p + q \): This is a rational number because the sum of two rational numbers is rational.
- \( p - q \): This is also a rational number since the difference of two rational numbers is rational.
- \( p \cdot q \): The product of two rational numbers is rational.
- \( \frac{p}{q} \) (assuming \( q \neq 0 \)): The quotient of two rational numbers is rational.
- \( p + r \): This is an irrational number because the sum of a rational number and an irrational number is irrational.
- \( p - r \): This is also an irrational number for the same reason as above.
- \( p \cdot r \): This is an irrational number, as the product of a rational number and an irrational number remains irrational.
- \( \frac{p}{r} \) (assuming \( r \neq 0 \)): This is an irrational number as well.
From the above operations, the expressions that represent rational numbers are:
- \( p + q \)
- \( p - q \)
- \( p \cdot q \)
- \( \frac{p}{q} \) (for \( q \neq 0 \))
Any combination of \( p \) and \( q \) in addition, subtraction, multiplication, or division (except division by zero) will yield a rational number. Conversely, any expression involving \( r \) combined with \( p \) or \( q \) will result in an irrational number.
Thus, expressions like \( p + q \) or \( p \cdot q \) represent rational numbers.