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.

Question

Expression
x+y

y+z

xy

xz

Answer 1

Let's evaluate each expression in terms of whether it is rational or irrational, given that \( x \) and \( y \) are non-zero, rational numbers, and \( z \) is an irrational number.

1. Expression: \( x + y \)

   • Since both \( x \) and \( y \) are rational numbers, their sum \( x + y \) is also a rational number.
   • Value: Rational

2. 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.
   • Value: Irrational

3. Expression: \( xy \)

   • The product of two rational numbers \( x \) and \( y \) is also a rational number.
   • Value: Rational

4. Expression: \( xz \)

   • Here, \( x \) is a rational number and \( z \) is an irrational number. The product of a rational number and an irrational number is always irrational.
   • Value: Irrational

Thus, the final classification of the values is:

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