Which statement is true about the product of a non-zero rational number and an irrational number?

Question

Responses
A The product of a non-zero rational number and an irrational number is always a rational number.The product of a non-zero rational number and an irrational number is always a rational number.
B The product of a non-zero rational number and an irrational number is never an irrational number.The product of a non-zero rational number and an irrational number is never an irrational number.
C The product of a non-zero rational number and an irrational number is sometimes a rational number.The product of a non-zero rational number and an irrational number is sometimes a rational number.
D The product of a non-zero rational number and an irrational number is always an irrational number.

Answer 1

The correct statement is:

D. The product of a non-zero rational number and an irrational number is always an irrational number.

Here's why:

1. A rational number can be expressed as a fraction (a/b) where a and b are integers, and b is not zero.
2. An irrational number cannot be expressed as a fraction of integers and has a non-repeating, non-terminating decimal expansion.
3. When you multiply a non-zero rational number with an irrational number, the result cannot be simplified to a rational form. Therefore, the product remains irrational.

Thus, the product of a non-zero rational number and an irrational number is always irrational.