Question
Suppose that p & q are nonzero rational numbers and z is an irrational number. Which of the following would be rational? Select TWO answers.
the sum of q and z
the product of p and q
the product of p and z
the sum of p and q
the difference of p and z
the sum of q and z
the product of p and q
the product of p and z
the sum of p and q
the difference of p and z
Answers
GPT-4o mini
To determine which of the options will result in a rational number, we will analyze each one:
1. **The sum of q and z (q + z)**: Since \( q \) is rational and \( z \) is irrational, their sum will be irrational. Thus, this is **not rational**.
2. **The product of p and q (p * q)**: Since both \( p \) and \( q \) are nonzero rational numbers, their product will also be a rational number. Thus, this **is rational**.
3. **The product of p and z (p * z)**: Here, \( p \) is rational and \( z \) is irrational. The product of a rational number and an irrational number is irrational. Thus, this is **not rational**.
4. **The sum of p and q (p + q)**: Since both \( p \) and \( q \) are rational, their sum will also be a rational number. Thus, this **is rational**.
5. **The difference of p and z (p - z)**: Similar to the product case, since \( p \) is rational and \( z \) is irrational, their difference will be irrational. Thus, this is **not rational**.
After evaluating all options, the two answers that are rational are:
- The product of p and q (p * q)
- The sum of p and q (p + q)
1. **The sum of q and z (q + z)**: Since \( q \) is rational and \( z \) is irrational, their sum will be irrational. Thus, this is **not rational**.
2. **The product of p and q (p * q)**: Since both \( p \) and \( q \) are nonzero rational numbers, their product will also be a rational number. Thus, this **is rational**.
3. **The product of p and z (p * z)**: Here, \( p \) is rational and \( z \) is irrational. The product of a rational number and an irrational number is irrational. Thus, this is **not rational**.
4. **The sum of p and q (p + q)**: Since both \( p \) and \( q \) are rational, their sum will also be a rational number. Thus, this **is rational**.
5. **The difference of p and z (p - z)**: Similar to the product case, since \( p \) is rational and \( z \) is irrational, their difference will be irrational. Thus, this is **not rational**.
After evaluating all options, the two answers that are rational are:
- The product of p and q (p * q)
- The sum of p and q (p + q)