Problem 13 - Prealgebra and Introductory Algebra

Question

Multiply. $$-8(2)$$

Step-by-Step Solution

Answer

The result of the multiplication \(-8 * 2\) is \(-16\).

Step 1: Identify the numbers to multiply

The numbers to be multiplied are \(-8\) and \(2\). The negative symbol indicates that the result will also be negative.

Step 2: Perform the multiplication

Multiply the absolute values of \(-8\) and \(2\), which gives \(16\).

Step 3: Attach the correct sign to the result

A negative number (\(-8\)) multiplied by a positive number (\(2\)) results in a negative number. Therefore, the result of the multiplication \(-8 * 2\) is \(-16\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They're commonly used to signify values below a baseline, such as temperature or owing money. When multiplying negative numbers by positive numbers, it’s important to remember that:

• A negative times a positive results in a negative.
• A negative times a negative results in a positive.
• A positive times a positive results in a positive.

In our exercise, multiplying \(-8\) (negative) by \(2\) (positive) gives us \(-16\). This is because a negative multiplied by a positive always results in a negative. Understanding this property helps you determine the sign of your answer quickly.

Working with Absolute Values

Absolute values represent the distance of a number from zero on a number line, regardless of direction. They are always non-negative. For example, the absolute value of \(-8\) is \(8\), and the absolute value of \(2\) is \(2\). The notation for absolute value is \(|x|\), where \(x\) is any number.

• \(|-8| = 8\)
• \(|2| = 2\)
• \(|-16| = 16\)

When multiplying, first consider the absolute values of the numbers. For our problem:\[|-8| \times |2| = 8 \times 2 = 16\]Then apply the rules for negative and positive signs from our understanding of negative numbers. The absolute value helps simplify multiplication by focusing on magnitude.

Exploring Multiplication Properties

Multiplication properties are fundamental principles that help in performing multiplication efficiently and correctly. Three key properties include:

• Commutative Property: The order of multiplication doesn’t affect the product. For any numbers \(a\) and \(b\), \(a \times b = b \times a\).
• Associative Property: When multiplying more than two numbers, the grouping doesn’t impact the result. \((a \times b) \times c = a \times (b \times c)\).
• Identity Property: Any number multiplied by \(1\) remains unchanged. \(a \times 1 = a\).

In our example with \(-8\) and \(2\), we particularly see the interaction of signs which is guided by these properties and by our understanding of multiplying negative and positive numbers.