How Is The Sign Of $18(-24)(-5)(-101)(-7)$ Determined? Complete The Statement.The Product Is $\square$ Because There Is An $\square$ Number Of Negative Factors.

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Introduction

When it comes to multiplying numbers, the sign of the product is determined by the signs of the individual factors. In this article, we will delve into the world of negative numbers and explore how the sign of a product is determined. We will examine the properties of negative numbers, the rules for multiplying them, and provide examples to illustrate the concepts.

The Properties of Negative Numbers

A negative number is a number that is less than zero. It is denoted by a minus sign (-) and is the opposite of a positive number. For example, -3 is a negative number, and it is the opposite of 3.

The Rules for Multiplying Negative Numbers

When multiplying two or more numbers, the sign of the product is determined by the signs of the individual factors. There are two rules to remember:

  • An even number of negative factors results in a positive product: If there are an even number of negative factors, the product will be positive.
  • An odd number of negative factors results in a negative product: If there are an odd number of negative factors, the product will be negative.

Determining the Sign of a Product

To determine the sign of a product, we need to count the number of negative factors. If the number of negative factors is even, the product will be positive. If the number of negative factors is odd, the product will be negative.

Example 1: An Even Number of Negative Factors

Let's consider the product of -2, -3, and 4. We have three factors, and two of them are negative. Since there are an even number of negative factors, the product will be positive.

(−2)(−3)(4)=24(-2)(-3)(4) = 24

In this example, the product is 24, which is a positive number.

Example 2: An Odd Number of Negative Factors

Now, let's consider the product of -2, -3, and -4. We have three factors, and all of them are negative. Since there are an odd number of negative factors, the product will be negative.

(−2)(−3)(−4)=−24(-2)(-3)(-4) = -24

In this example, the product is -24, which is a negative number.

Example 3: A Mixed Product

Let's consider the product of -2, 3, and -4. We have three factors, and two of them are negative. Since there are an even number of negative factors, the product will be positive.

(−2)(3)(−4)=24(-2)(3)(-4) = 24

In this example, the product is 24, which is a positive number.

Conclusion

In conclusion, the sign of a product is determined by the signs of the individual factors. If there are an even number of negative factors, the product will be positive. If there are an odd number of negative factors, the product will be negative. By understanding the properties of negative numbers and the rules for multiplying them, we can determine the sign of a product with ease.

Practice Problems

  1. Determine the sign of the product of -5, -2, and 3.
  2. Determine the sign of the product of -3, -4, and -5.
  3. Determine the sign of the product of -2, 3, and -4.

Answers

  1. The product is positive because there are an even number of negative factors.
  2. The product is negative because there are an odd number of negative factors.
  3. The product is positive because there are an even number of negative factors.

Final Thoughts

Q: What is the rule for determining the sign of a product?

A: The rule for determining the sign of a product is as follows:

  • An even number of negative factors results in a positive product: If there are an even number of negative factors, the product will be positive.
  • An odd number of negative factors results in a negative product: If there are an odd number of negative factors, the product will be negative.

Q: How do I determine the number of negative factors in a product?

A: To determine the number of negative factors in a product, you need to count the number of negative numbers being multiplied together. For example, if you have the product of -2, -3, and 4, you would count the number of negative factors as follows:

  • -2 is a negative factor
  • -3 is a negative factor
  • 4 is a positive factor

In this case, there are 2 negative factors, so the product will be positive.

Q: What if I have a mixed product with both positive and negative factors?

A: If you have a mixed product with both positive and negative factors, you need to count the number of negative factors to determine the sign of the product. For example, if you have the product of -2, 3, and -4, you would count the number of negative factors as follows:

  • -2 is a negative factor
  • 3 is a positive factor
  • -4 is a negative factor

In this case, there are 2 negative factors, so the product will be positive.

Q: Can you provide more examples of determining the sign of a product?

A: Here are a few more examples:

  • Example 1: Determine the sign of the product of -5, -2, and 3.
    • -5 is a negative factor
    • -2 is a negative factor
    • 3 is a positive factor
    • There are 2 negative factors, so the product will be positive.
  • Example 2: Determine the sign of the product of -3, -4, and -5.
    • -3 is a negative factor
    • -4 is a negative factor
    • -5 is a negative factor
    • There are 3 negative factors, so the product will be negative.
  • Example 3: Determine the sign of the product of -2, 3, and -4.
    • -2 is a negative factor
    • 3 is a positive factor
    • -4 is a negative factor
    • There are 2 negative factors, so the product will be positive.

Q: How do I apply the rules for determining the sign of a product to real-world problems?

A: The rules for determining the sign of a product can be applied to a wide range of real-world problems, such as:

  • Physics: When calculating the force of a moving object, you need to consider the signs of the forces acting on the object.
  • Engineering: When designing a system, you need to consider the signs of the forces and energies involved.
  • Economics: When analyzing the impact of a policy change, you need to consider the signs of the economic indicators.

By applying the rules for determining the sign of a product, you can ensure that your calculations are accurate and reliable.

Q: What are some common mistakes to avoid when determining the sign of a product?

A: Here are a few common mistakes to avoid:

  • Not counting the number of negative factors correctly: Make sure to count the number of negative factors carefully to avoid mistakes.
  • Not considering the signs of the factors: Make sure to consider the signs of all the factors in the product.
  • Not applying the rules consistently: Make sure to apply the rules consistently to avoid mistakes.

By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.