Consider the sums and products of nonzero rational numbers and irrational numbers.

Question

Move the options to the spaces to show whether each statement is always true, sometimes true, or never true.

Response area with 4 blank spaces
Statement
Always,

Sometimes, or

Never True

The sum of two nonzero rational numbers is rational.
Blank space 4 empty
​

The sum of a nonzero rational number and an irrational number is rational.
Blank space 5 empty

The product of two irrational numbers is irrational.
Blank space 6 empty

The product of a nonzero rational number and an irrational number is rational.
Blank space 7 empty
​


Answer options with 3 options.

Answer 1

Here are the statements categorized into the appropriate spaces:

1. The sum of two nonzero rational numbers is rational.

   • Always True

2. The sum of a nonzero rational number and an irrational number is rational.

   • Never True

3. The product of two irrational numbers is irrational.

   • Sometimes True (e.g., the product of √2 and √2 is 2, which is rational, but the product of √2 and √3 is irrational)

4. The product of a nonzero rational number and an irrational number is rational.

   • Never True (e.g., rational number times an irrational number is always irrational)

So, the final categorization looks like this:

• Blank space 4 (The sum of two nonzero rational numbers is rational): Always True
• Blank space 5 (The sum of a nonzero rational number and an irrational number is rational): Never True
• Blank space 6 (The product of two irrational numbers is irrational): Sometimes True
• Blank space 7 (The product of a nonzero rational number and an irrational number is rational): Never True