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Chapter 1: Problem 14

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

Short Answer

Expert verified
The product is 12.

Step by step solution

01

Identify the Problem

Given the expression \(-3(-4)\), we need to find the product of the two numbers.
02

Recall the Rule for Multiplying Two Negative Numbers

When multiplying two negative numbers, the product is positive. This means \(-a(-b) = ab\).
03

Apply the Rule

Using the rule for multiplying two negative numbers, multiply \(-3\) by \(-4\). This gives \(-3 \times -4 = 12\).
04

Write the Final Answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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\.

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Most popular questions from this chapter

Use the distributive property to rewrite each expression. $$ 3(3 x+4) $$

Write a numerical expression for each phrase, and simplify the expression. 1.85 more than the sum of -1.25 and -4.75

Use the distributive property to rewrite each expression. $$ 2(6 x+5) $$

Write a numerical expression for each phrase, and simplify the expression. 19 less than the difference of 9 and -2

Write each sentence as an equation, using \(x\) as the variable. Then find the solution of the equation from the set of integers between -12 and \(12,\) inclusive. An integer is divisible by 12 if it is divisible by both 3 and \(4,\) and not otherwise. (a) Is 4,249,474 divisible by \(12 ?\) (b) Is 4,253,520 divisible by \(12 ?\)
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