Problem 56 - Algebra 1: Concepts and Skills

Question

Find the product. $$ -8(-5) $$

Step-by-Step Solution

Answer

The product of -8 and -5 is 40.

Step 1: Identify the Numbers to Be Multiplied

The exercise provides two numbers to be multiplied, which are -8 and -5.

Step 2: Apply the Rules of Multiplication for Negative Numbers

Recall that the product of two negative numbers is a positive number. Hence, \((-8) \times (-5)\) will yield a positive number.

Step 3: Perform the Multiplication

Perform the multiplication of the absolute values of the numbers: \(8 \times 5\) equals \(40\). Since both numbers are negative, the result will be positive, so the final answer is \(40\).

Key Concepts

Understanding Negative Numbers

Negative numbers are simply numbers less than zero. They are usually used to represent a loss or a value below a baseline, like below sea level or withdrawal from a bank account.
Some important facts about negative numbers include:

• They are represented with a minus sign, such as -1, -2, and so on.
• On the number line, negative numbers are to the left of zero.
• The farther left they are on the number line, the smaller their value.

One core principle in using negative numbers is that multiplying them can turn them into positive values, under certain conditions.

Rules of Multiplication for Negative Numbers

Whenever you multiply two numbers, their signs will affect the outcome. Here are the rules you should remember about multiplying negative numbers:

• If you multiply two positive numbers, the result is positive. For example, \(3 \times 4 = 12\).
• If you multiply two negative numbers, the result is positive as well. For example, when we have \(-8 \times -5\), we end up with a positive \(40\).
• If one number is negative and the other is positive, the result will be negative. An example would be \(-8 \times 5 = -40\).

These rules help us predict whether the product of two numbers will be positive or negative, simply by looking at their signs.

Understanding Absolute Value

When dealing with any number, the absolute value represents its distance from zero on the number line, without considering which side of zero it is on.
Some key points regarding absolute value:

• The absolute value of a positive number is the number itself. For example, the absolute value of \(8\) is \(8\).
• The absolute value of a negative number is its positive counterpart. Thus, the absolute value of \(-8\) is \(8\).
• Absolute values are always non-negative.

In the multiplication of negative numbers, as shown in the step-by-step solution, after recognizing the signs, we focus on the absolute values to easily perform the multiplication: \(8 \times 5 = 40\), and since negative times negative gives a positive result, the product is a positive \(40\). This makes calculations intuitive and hassle-free.