The product of a non-zero rational number with an irrational number is always a/an

Options

A. always irrational
B. always rational
C. rational or irrational
D. one

Explanation

To solve the question, we need to determine the nature of the product of a non-zero rational number and an irrational number.

### Step-by-Step Solution:

1. **Understanding Rational and Irrational Numbers**:
- A rational number is a number that can be expressed as the quotient of two integers (e.g., 1/2, 3, -4).
- An irrational number is a number that cannot be expressed as a simple fraction (e.g., √2, π).

2. **Choosing a Non-Zero Rational Number**:
- Let's select a simple non-zero rational number. For example, we can choose \(2\).

3. **Choosing an Irrational Number**:
- Now, we need to choose an irrational number. A common example is \( \sqrt{3} \).

4. **Calculating the Product**:
- We will multiply the chosen rational number by the irrational number:
\[ 2 \times \sqrt{3} = 2\sqrt{3} \]

5. **Determining the Nature of the Product**:
- Now, we need to determine if \(2\sqrt{3}\) is rational or irrational.
- Since \( \sqrt{3} \) is irrational, multiplying it by a non-zero rational number (2 in this case) will still yield an irrational number.

6. **Conclusion**:
- Therefore, the product of a non-zero rational number and an irrational number is always an irrational number.

### Final Answer:
The product of a non-zero rational number with an irrational number is always an **irrational number**.