Question
Consider the sums and products of nonzero rational numbers and irrational numbers.
Move the options to the spaces to show whether each statement is always true, sometimes true, or never true.
Statement
Always,
Sometimes, or
Never True
The sum of two nonzero rational numbers is rational.
The sum of a nonzero rational number and an irrational number is rational.
The product of two irrational numbers is irrational.
The product of a nonzero rational number and an irrational number is rational.
sometimes true
always true
Move the options to the spaces to show whether each statement is always true, sometimes true, or never true.
Statement
Always,
Sometimes, or
Never True
The sum of two nonzero rational numbers is rational.
The sum of a nonzero rational number and an irrational number is rational.
The product of two irrational numbers is irrational.
The product of a nonzero rational number and an irrational number is rational.
sometimes true
always true
Answers
Answered by
GPT-4o mini
Let's evaluate each statement regarding the sums and products of nonzero rational numbers and irrational numbers.
1. **The sum of two nonzero rational numbers is rational.**
- **Always True**: The sum of two rational numbers is always rational.
2. **The sum of a nonzero rational number and an irrational number is rational.**
- **Never True**: The sum of a rational number and an irrational number is always irrational.
3. **The product of two irrational numbers is irrational.**
- **Sometimes True**: The product of two irrational numbers can be rational (for example, \(\sqrt{2} \times \sqrt{2} = 2\)), but it is also often irrational (e.g., \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\)).
4. **The product of a nonzero rational number and an irrational number is rational.**
- **Never True**: The product of a nonzero rational number and an irrational number is always irrational (for example, \(2 \times \sqrt{2} = 2\sqrt{2}\) which is irrational).
So the correct categorization is as follows:
- **The sum of two nonzero rational numbers is rational.** → Always True
- **The sum of a nonzero rational number and an irrational number is rational.** → Never True
- **The product of two irrational numbers is irrational.** → Sometimes True
- **The product of a nonzero rational number and an irrational number is rational.** → Never True
1. **The sum of two nonzero rational numbers is rational.**
- **Always True**: The sum of two rational numbers is always rational.
2. **The sum of a nonzero rational number and an irrational number is rational.**
- **Never True**: The sum of a rational number and an irrational number is always irrational.
3. **The product of two irrational numbers is irrational.**
- **Sometimes True**: The product of two irrational numbers can be rational (for example, \(\sqrt{2} \times \sqrt{2} = 2\)), but it is also often irrational (e.g., \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\)).
4. **The product of a nonzero rational number and an irrational number is rational.**
- **Never True**: The product of a nonzero rational number and an irrational number is always irrational (for example, \(2 \times \sqrt{2} = 2\sqrt{2}\) which is irrational).
So the correct categorization is as follows:
- **The sum of two nonzero rational numbers is rational.** → Always True
- **The sum of a nonzero rational number and an irrational number is rational.** → Never True
- **The product of two irrational numbers is irrational.** → Sometimes True
- **The product of a nonzero rational number and an irrational number is rational.** → Never True