Problem 6 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-4(-7)$$

Step-by-Step Solution

Answer

The product of \(-4\) and \(-7\) is 28.

Step 1: Understand the Problem

We need to multiply two negative numbers: \(-4\) and \(-7\). The multiplication of two numbers with the same sign will always result in a positive number.

Step 2: Multiply the Absolute Values

First, we consider the absolute values of the numbers, which are 4 and 7. Multiply these absolute values: \(4 \times 7 = 28\).

Step 3: Determine the Sign of the Product

Since both numbers were negative, the product of two negative numbers is positive. Therefore, the sign of the product is positive.

Step 4: Write the Final Product

Combine the results from the previous steps to give a positive product from Step 2. Thus, \(-4 \times -7 = 28\).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering which direction from zero the number is. Whether a number is positive or negative, its absolute value is always non-negative. For example:

• The absolute value of \(-4\) is 4.
• The absolute value of \(-7\) is 7.

This is important in multiplication because you first ignore the negative signs and multiply the absolute values. Once you have the absolute value of the product, you then determine the sign of the result based on the rules of multiplying integers.

Product of Integers

When multiplying integers, it's crucial to understand the concept of a product. A product is the result you get after multiplying two or more numbers. In the case of multiplying \(-4\) and \(-7\), you look at their absolute values first, as noted in step 2:

• The product of the absolute values 4 and 7 is calculated as follows:
• \[ 4 \times 7 = 28 \]

The raw number you get from multiplying the absolute values is necessary so you can later apply the sign rule. This step separates the operation into two parts: working with magnitude first and then with sign.

Integer Multiplication

Multiplying integers involves not just calculating the magnitude of the product but also determining its sign. When you have both numbers with the same sign, whether positive or negative, the product will always end up being positive.

• If both integers are negative, the negative signs "cancel out," ensuring the result is positive.
• For example, the multiplication \(-4\times -7\) results in a positive 28 because negative \(\times\) negative equals positive.

Conversely, if one integer is positive and the other is negative, the product will be negative. This sign rule is fundamental in understanding integer multiplication correctly. In this exercise, since both integers are negative, we follow the rule and conclude the product is positive, resulting in \(28\).