The product of two positive integers and eight negative integers is a

Answer

To solve the problem, we need to determine the sign of the product of two positive integers and eight negative integers.

### Step-by-Step Solution:

1. **Identify the integers**:
- We have 2 positive integers (let's denote them as \( a \) and \( b \)).
- We have 8 negative integers (let's denote them as \( -c_1, -c_2, -c_3, -c_4, -c_5, -c_6, -c_7, -c_8 \)).

2. **Calculate the product of the positive integers**:
- The product of the two positive integers is \( a \times b \).
- Since both \( a \) and \( b \) are positive, \( a \times b \) is also positive.

3. **Calculate the product of the negative integers**:
- The product of the 8 negative integers can be expressed as:
\[ (-c_1) \times (-c_2) \times (-c_3) \times (-c_4) \times (-c_5) \times (-c_6) \times (-c_7) \times (-c_8) \]
- When multiplying two negative numbers, the result is positive. Therefore, we can pair the negative integers:
- \( (-c_1) \times (-c_2) \) is positive.
- \( (-c_3) \times (-c_4) \) is positive.
- \( (-c_5) \times (-c_6) \) is positive.
- \( (-c_7) \times (-c_8) \) is positive.
- Since there are 8 negative integers, we can group them into 4 pairs, resulting in a positive product.

4. **Determine the overall product**:
- Now, we have:
\[ \text{Positive product from positive integers} \times \text{Positive product from negative integers} \]
- Since both products are positive, the overall product is positive.

5. **Conclusion**:
- Therefore, the product of 2 positive integers and 8 negative integers is a positive number.

### Final Answer:
The product of two positive integers and eight negative integers is a positive number.

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