What can you say about the product of an non zero rational and irrational number?

Answer

To determine the nature of the product of a non-zero rational number and an irrational number, we can follow these steps:

### Step-by-Step Solution:

1. **Define Rational and Irrational Numbers**:
- A rational number can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).
- An irrational number cannot be expressed as a fraction of integers. Examples include numbers like \( \sqrt{2} \), \( \pi \), etc.

2. **Choose a Non-Zero Rational Number**:
- Let's select a simple non-zero rational number. For example, let’s take \( 2 \).

3. **Choose an Irrational Number**:
- Now, we will choose an irrational number. A common choice is \( \sqrt{2} \).

4. **Calculate the Product**:
- Now, we will calculate the product of the chosen rational number and the irrational number:
\[ \text{Product} = 2 \times \sqrt{2} \]
- This simplifies to:
\[ 2\sqrt{2} \]

5. **Determine the Nature of the Result**:
- We need to determine whether \( 2\sqrt{2} \) is a rational or an irrational number.
- Since \( \sqrt{2} \) is irrational, any non-zero rational number multiplied by an irrational number results in an irrational number. Therefore, \( 2\sqrt{2} \) is an irrational number.

### Conclusion:
The product of a non-zero rational number and an irrational number is always an irrational number.