Problem 14 - Beginning and Intermediate Algebra

Question

Find each product. \(-3(-4)\)

Step-by-Step Solution

Answer

The product is 12.

Step 1: Identify the Problem

Given the expression \(-3(-4)\), we need to find the product of the two numbers.

Step 2: Recall the Rule for Multiplying Two Negative Numbers

When multiplying two negative numbers, the product is positive. This means \(-a(-b) = ab\).

Step 3: Apply the Rule

Using the rule for multiplying two negative numbers, multiply \(-3\) by \(-4\). This gives \(-3 \times -4 = 12\).

Step 4: Write the Final Answer

The product of \(-3\) and \(-4\) is 12.

Key Concepts

Product of Integers

Understanding how to find the product of integers, especially when they involve negative numbers, is crucial in mathematics. An integer is any whole number, either positive or negative, including zero.

When we talk about the product of integers, we mean the result you get when you multiply two or more integers together. For example, in the exercise \(-3(-4)\), we need to find the multiplication result.

The positive or negative nature of the integers affects the final product. Negative integers indicate the opposite direction on the number line compared to positive integers. Let's delve deeper into the rules that govern these multiplications.

Multiplication Rules

Multiplying positive and negative integers follows specific rules. Understanding these rules ensures you get the correct product. Here are the rules when dealing with these numbers:

• A positive number multiplied by a positive number results in a positive product. \(4 \times 3 = 12\).
• A positive number multiplied by a negative number results in a negative product. \(4 \times (-3) = -12\).
• A negative number multiplied by a negative number results in a positive product. For example, \(-3 \times -4 = 12\). This is because the two negatives cancel each other out, making the product positive.

These rules are in place because of how multiplication impacts the direction on the number line. Negative times negative gives positive as it changes direction twice.

Negative Numbers

Negative numbers are less than zero and are typically represented with a minus sign in front of them. Understanding negative numbers is essential for correctly applying multiplication rules.

When you multiply two negative numbers, think of it as reversing a direction twice. Individually, each negative number changes the direction on the number line once. When multiplied together, the two reversals bring you back to a positive direction. Thus, the product is positive.

Let's look again at our example \( -3(-4)\). You treat the negative signs as indicators of direction change. Hence, multiplying \(-3\) by \(-4\) ultimately leaves you with a positive \12\.