Problem 12 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-8(-3)$$

Step-by-Step Solution

Answer

The product of \(-8 \times -3\) is 24.

Step 1: Understand the Problem

We need to find the product of two negative numbers, specifically \(-8(-3)\). This exercise tests our understanding of how multiplying two negative numbers works.

Step 2: Apply the Rule of Negative Multiplication

When multiplying two negative numbers, the result is positive. This is because a negative number times a negative number equals a positive number. Therefore, \(-8 \times -3\ = +(8 \times 3)\).

Step 3: Calculate the Product

Calculate the product of the absolute values: \(8 \times 3 = 24\). Since both original numbers were negative, we know the final result is positive. Therefore, \(-8 \times -3 = 24\).

Key Concepts

Negative Multiplication Rule

Multiplying negative numbers might seem tricky at first, but there's a simple rule to remember: when you multiply two negative numbers, the result is positive. This phenomenon can be understood by thinking about the definition of multiplication as repeated addition.

For example, consider the product \(-8 imes -3\). We can understand this by thinking about subtraction. Negative signs essentially mean "opposite," so \(-8\) means the opposite of 8. When you multiply two negatives, it's like reversing direction twice, which brings you back to the positive.

In simpler terms, think of it this way: if turning right is negative and you turn right twice, you're actually back to facing forward, the positive direction. Hence, \(-8 imes -3\) gives a positive 24.

Absolute Value in Multiplication

Understanding absolute values is crucial in multiplication, especially when negative numbers are involved. Absolute value refers to the distance a number is from zero on the number line, ignoring direction.

For example, the absolute value of \(-8\) is \(|-8| = 8\), and the absolute value of \(-3\) is \(|-3| = 3\). When multiplying, focus on these absolute values first.

In our exercise, after stripping away the negative signs, we multiply 8 and 3 to get 24. We then apply the negative multiplication rule, which tells us that two negatives make a positive, finalizing our product at 24.

Product of Integers

When multiplying integers, the process follows specific and predictable rules, whether they are positive or negative. Integers are whole numbers, both positive and negative, including zero.

To find the product of integers, you multiply the absolute values of the numbers first. For example, in \(-8 imes -3\), you consider the absolute values and multiply 8 and 3 to get 24.

Next, apply the sign rules. Two negatives create a positive, so our final product is positive. If one number were positive and the other negative, the product would be negative.

• Two positives make a positive product.
• Two negatives make a positive product.
• A positive and a negative make a negative product.

Understanding these simple rules makes multiplying integers straightforward, regardless of their signs.