Answer :
To determine the sign of the product when multiplying several integers, you can use a simple rule: the product of an odd number of negative integers is negative, and the product of an even number of negative integers is positive. This is because each pair of negative numbers multiplied together gives a positive result.
Here's a step-by-step explanation:
Identify the signs of the integers involved:
- You have 10 positive integers. Each of these contributes a positive factor to the product.
- You have 197 negative integers. Each of these contributes a negative factor to the product.
Determine the total sign contributions:
- Positive integers will not change the overall sign. Multiplying any number with a positive integer keeps the product's sign unchanged.
- Negative integers contribute to flipping the sign; two negative numbers result in a positive product, so we focus on pairing them.
Count the number of negative integers:
- There are 197 negative integers.
- Since 197 is an odd number, an odd number of negative integers will result in a negative product.
Combine the understanding:
- The 10 positive integers do not affect the overall negativity or positivity of the product.
- The odd number of negative integers (197) determines the product's sign.
Therefore, the sign of the product will be negative. In summary, multiplying 10 positive integers and 197 negative integers results in a negative product.
