Problem 30 - Beginning and Intermediate Algebra

Question

Multiply. $$ -12(12) $$

Step-by-Step Solution

Answer

The product is -144.

Step 1: Identify the Numbers

First, let's identify the numbers in the expression. We have -12 and 12.

Step 2: Understand the Multiplication of Signs

Recall the rule for multiplying two numbers with signs: a negative number times a positive number equals a negative number. Thus, the product will be negative.

Step 3: Multiply the Absolute Values

Now, multiply the absolute values of the numbers: \( 12 \times 12 \). This equals 144.

Step 4: Apply the Sign Rule

Since we are multiplying a negative number by a positive number, the result is negative. Therefore, the product is \( -144 \).

Key Concepts

Negative Numbers

Negative numbers are simply numbers that have a value less than zero. This is indicated by the minus sign (-) in front of the number. They are commonly used to represent a decrease in amounts, temperatures below zero, or financial debts. Unlike positive numbers, negative numbers run in the opposite direction on the number line.
They are crucial in various calculations because they help us understand phenomena that involve loss, deficiency, or direction changes, like moving backwards. Negative numbers are found in practically every aspect of math, from algebra to calculus, as they expand the possibilities of solutions and solutions sets.

Multiplying Integers

Multiplying integers involves combining groups of numbers together, including both positive and negative values. Integers are whole numbers that can be either positive, negative, or zero. When multiplying integers, you're essentially adding a number to itself a particular number of times.
For example, multiplying 3 by 4 ( 3 imes 4 ) means adding three instances of the number 4 together, resulting in 12. The process is done similarly for negative numbers. However, it's important to keep a close eye on the signs because they affect the final product.
An important tip to remember is that multiplying any number by zero results in zero, since you're essentially adding zero a number of times.

Sign Rules in Multiplication

The sign rules in multiplication help us determine the sign of the product when dealing with positive and negative integers. Understanding these rules is crucial to correctly performing operations with integers. There are three main sign rules:

• When a positive number is multiplied by a positive number, the result is positive.
• When a negative number is multiplied by a positive number, the product is negative. This is because you are essentially subtracting multiple groups of the positive value.
• When you multiply two negative numbers, the result is positive. This is because two negatives make a positive in the case of multiplication - essentially flipping the direction twice brings you back forwards.

As seen in our original exercise, multiplying -12 imes 12 results in -144 because a negative times a positive yields a negative product. These rules form the foundation for solving problems with integers efficiently.