Let x and y be non-zero, rational numbers, and let z be an irrational number. determine whether the value of each expression is rational or irrational.

Question

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

Answer 1

Let's analyze each expression using the properties of rational and irrational numbers:

1. \(x + y\):

   • Since \(x\) and \(y\) are both non-zero rational numbers, the sum of two rational numbers is rational.
   • Result: Rational.

2. \(y + z\):

   • 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.
   • Result: Irrational.

3. \(xy\):

   • Since both \(x\) and \(y\) are non-zero rational numbers, the product of two rational numbers is also rational.
   • Result: Rational.

4. \(xz\):

   • 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 always irrational.
   • Result: Irrational.

In summary:

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