Question

Let x and y be non-zero, rational numbers, and let z be an irrational number. In the first column of the table below is an expression. In the second column, determine whether the value of each expression is rational or irrational.(4 points)
Rational Irrational
y+z

xy

xz

x+y

Answer 1

To classify each of the expressions as rational or irrational, let's analyze them one by one:

1. **y + z**
- Here, \(y\) is a non-zero rational number, and \(z\) is an irrational number.
- The sum of a rational number and an irrational number is always irrational.
- **Classification**: Irrational

2. **xy**
- Both \(x\) and \(y\) are non-zero rational numbers.
- The product of two rational numbers is always rational.
- **Classification**: Rational

3. **xz**
- Here, \(x\) is a non-zero rational number, and \(z\) is an irrational number.
- The product of a rational number and an irrational number is always irrational.
- **Classification**: Irrational

4. **x + y**
- Both \(x\) and \(y\) are non-zero rational numbers.
- The sum of two rational numbers is always rational.
- **Classification**: Rational

Now, summarizing the classifications:

| Expression | Classification |
|------------|----------------|
| y + z | Irrational |
| xy | Rational |
| xz | Irrational |
| x + y | Rational |