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.

1. xz
2. x+y
3. xy
4. y+z

Answer 1

Let's analyze each expression to determine whether its value is rational or irrational.

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

2. **x+y**: Both \( x \) and \( y \) are non-zero rational numbers. The sum of two rational numbers is always rational. Therefore, **x+y is rational**.

3. **xy**: Again, both \( x \) and \( y \) are non-zero rational numbers. The product of two rational numbers is always rational. Therefore, **xy is rational**.

4. **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. Therefore, **y+z is irrational**.

To summarize:

1. \( xz \) - **Irrational**
2. \( x+y \) - **Rational**
3. \( xy \) - **Rational**
4. \( y+z \) - **Irrational**