Question

If you multiply four negative numbers, what is the sign of the product? If you multiply five negative numbers, what is the sign of the product? Based on the answers to these questions and the product rules, write a general rule for the sign of the product based on the number of negative numbers in the expression.

Answer 1

When multiplying negative numbers, the sign of the product depends on the number of negative numbers involved:

1. **Four negative numbers**: When you multiply four negative numbers, you are essentially multiplying pairs of negative numbers:
- The product of two negative numbers is positive (e.g., \((-a) \times (-b) = ab\)).
- Therefore, the product of the first two negative numbers is positive, and the product of the next two negative numbers is also positive. When you multiply these two positive results together, the final product is positive.
- So, the product of four negative numbers is **positive**.

2. **Five negative numbers**: When you multiply five negative numbers, you would again group them:
- The first four negative numbers can be paired to yield a positive product (as explained above).
- Now you have the positive product from the first four negative numbers multiplied by the fifth negative number, which results in a negative product since multiplying a positive number by a negative number yields a negative number.
- So, the product of five negative numbers is **negative**.

### General Rule:
From the above observations, we can derive a general rule for the sign of the product based on the number of negative numbers \( n \):
- If \( n \) (the number of negative numbers) is even, the product is **positive**.
- If \( n \) is odd, the product is **negative**.

Thus, the sign of the product is determined by whether the count of negative numbers is even or odd.