Problem 13 - Elementary and Intermediate Algebra

Question

Multiply. $$ -8 \cdot 7 $$

Step-by-Step Solution

Answer

-56

Step 1: Understand the Problem

We need to multiply two numbers: \(-8\) and \(7\).

Step 2: Multiplication Rule

Remember the rule for multiplication involving negative numbers: A negative times a positive equals a negative.

Step 3: Perform the Multiplication

Multiply the absolute values of the numbers: \( | -8 | = 8 \) and \( | 7 | = 7 \), so: \( 8 \times 7 = 56 \).

Step 4: Apply the Sign Rule

Since one number is negative and the other is positive, the result will be negative: \( -8 \times 7 = -56 \).

Key Concepts

Negative Numbers

In mathematics, negative numbers are those numbers that are less than zero. They are typically presented with a minus sign, like \(-8\). Negative numbers are used to represent values below a defined point, like temperatures below freezing or debts below zero dollars.
Understanding negative numbers is crucial because they follow unique rules, especially in multiplication and addition.
Examples:

• \(-5\)
• \(-23\)
• \(-100\)

Remember, when you see a negative sign, it indicates a value less than zero.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number. For instance, the absolute value of \(-8\) is 8, written as \(|-8| = 8\). Similarly, the absolute value of 7 is 7, written as \(|7| = 7\).
The absolute value helps simplify calculations, especially when dealing with negative numbers. It removes the negative sign and lets you focus on the magnitude of the number.
Application:

• For \(-8\), \(| -8 | = 8\)
• For 7, \(| 7 | = 7\)

Multiply and Sign Rule

The multiplication of integers follows specific rules when it comes to positive and negative numbers. The signs of the numbers you are multiplying determine the sign of the final result. Here are the basic rules:

• Positive \( \times \) Positive \( = \) Positive (e.g., \(3 \times 2 = 6\))
• Negative \( \times \) Positive \( = \) Negative (e.g., \(-3 \times 2 = -6\))
• Positive \( \times \) Negative \( = \) Negative (e.g., \(3 \times -2 = -6\))
• Negative \( \times \) Negative \( = \) Positive (e.g., \(-3 \times -2 = 6\))

In our original exercise where \(-8\) is multiplied by 7, we use these rules:
1. Multiply the absolute values: \(8 \times 7 = 56\)
2. Apply the sign rule: since one number is negative and the other is positive, the result is negative. So, \(-8 \times 7 = -56\).