Question

Suppose that p and q are rational numbers and r is an irrational number. Which expression represents a rational number?

Answer 1

give me the equation

Answer 2

To determine which expression represents a rational number among combinations of rational numbers \(p\) and \(q\) along with an irrational number \(r\), it is essential to remember the properties of rational and irrational numbers:

1. The sum or difference of two rational numbers is rational.
2. The product of two rational numbers is rational.
3. The sum or difference of a rational number and an irrational number is irrational.
4. The product of a rational number (other than zero) and an irrational number is irrational.
5. The quotient of a non-zero rational number and an irrational number is irrational.

Based on these properties, the following cases can be evaluated:

- **\(p + q\)**: This is rational since both \(p\) and \(q\) are rational.
- **\(p - q\)**: This is rational for the same reason.
- **\(p \times q\)**: This is also rational since both are rational.
- **\(p + r\)**: This is irrational, because the sum of a rational and irrational number is irrational.
- **\(p - r\)**: This is also irrational for the same reason.
- **\(p \times r\)**: This is irrational, as the product of a rational (other than zero) and an irrational number is irrational.
- **\(p / r\)**: This is irrational, as the quotient of a rational (non-zero) number and an irrational number is irrational.

So, the expressions that represent a rational number are:
- \(p + q\)
- \(p - q\)
- \(p \times q\)

Any expression that combines \(r\) with \(p\) or \(q\) in addition to the operations listed leads to an irrational result.

In conclusion, the expressions \(p + q\), \(p - q\), and \(p \times q\) are the ones that represent a rational number.

Answer 3

Here are some equations that represent rational numbers when \(p\) and \(q\) are rational numbers:

1. **Addition of two rational numbers:**
\[
r_1 = p + q
\]

2. **Subtraction of two rational numbers:**
\[
r_2 = p - q
\]

3. **Multiplication of two rational numbers:**
\[
r_3 = p \times q
\]

Any of these equations will yield a rational number, given that \(p\) and \(q\) are rational. If you need a specific equation with actual values for \(p\) and \(q\), please provide those values!