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
x+y


xy


y+z


xz

Answer 1

Let's analyze each expression one by one, given that \( x \) and \( y \) are non-zero rational numbers and \( z \) is an irrational number.

1. **Expression: \( x + y \)**
- Since \( x \) and \( y \) are both rational numbers, their sum \( x + y \) is also a rational number.
- **Conclusion: Rational**

2. **Expression: \( xy \)**
- The product of two rational numbers \( x \) and \( y \) is also a rational number.
- **Conclusion: Rational**

3. **Expression: \( y + z \)**
- Here, \( y \) is a rational number and \( z \) is an irrational number. The sum of a rational number and an irrational number is always irrational.
- **Conclusion: Irrational**

4. **Expression: \( xz \)**
- In this case, \( x \) is a rational number and \( z \) is an irrational number. The product of a rational number (non-zero) and an irrational number is irrational.
- **Conclusion: Irrational**

Here’s the summary of the conclusions for each expression:

| Expression | Type |
|------------|----------|
| \( x + y \) | Rational |
| \( xy \) | Rational |
| \( y + z \) | Irrational |
| \( xz \) | Irrational |