Question

suppose that p and q are rational numbers and r is an irrational number. which expression represents a ration number

Answer 1

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:

1. **\( p + q \)**: This is a rational number because the sum of two rational numbers is rational.
2. **\( p - q \)**: This is also a rational number since the difference of two rational numbers is rational.
3. **\( p \cdot q \)**: The product of two rational numbers is rational.
4. **\( \frac{p}{q} \)** (assuming \( q \neq 0 \)): The quotient of two rational numbers is rational.
5. **\( p + r \)**: This is an irrational number because the sum of a rational number and an irrational number is irrational.
6. **\( p - r \)**: This is also an irrational number for the same reason as above.
7. **\( p \cdot r \)**: This is an irrational number, as the product of a rational number and an irrational number remains irrational.
8. **\( \frac{p}{r} \)** (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.