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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 Numbers

Identify the numbers involved in the multiplication. In this case, the numbers are \( -3 \) and \( -4 \).
02

- Understand the Sign Rule in Multiplication

Recall the rule for multiplying two negative numbers: when two negative numbers are multiplied, the result is positive. Thus, \( -3 \times -4 \) will yield a positive result.
03

- Multiply the Absolute Values

Calculate the product of the absolute values of the numbers. The absolute value of \( -3 \) is \( 3 \), and the absolute value of \( -4 \) is \( 4 \). Multiply these together: \[ 3 \times 4 = 12 \]
04

- Apply the Sign Rule

Since the product of two negative numbers is positive, applying the sign rule to the previous result we get \( 12 \).

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on the number line, without considering direction. Essentially, it tells us how 'big' a number is, regardless of its sign.
For example:
  • The absolute value of -3 is 3, written as \(|-3| = 3\)
  • The absolute value of 4 is 4, written as \(|4| = 4\)
Absolute values are always non-negative, and this concept helps in simplifying multiplications and other operations.
In multiplication, when dealing with negative numbers, it's crucial to first understand and work with their absolute values. This neutralizes the effect of negative signs for the initial calculation stage.
Sign Rule in Multiplication
Multiplying numbers isn't just about combining their magnitudes; we also need to understand how their signs (positive or negative) affect the product.
Here are the key rules:
  • Positive \( \times \) Positive = Positive: \( 3 \times 4 = 12 \)
  • Negative \( \times \) Positive = Negative: \( -3 \times 4 = -12 \)
  • Positive \( \times \) Negative = Negative: \( 3 \times -4 = -12 \)
  • Negative \( \times \) Negative = Positive: \( -3 \times -4 = 12 \)
As seen from above, multiplying two negative numbers yields a positive product. This is because the two negatives 'cancel out' each other, in a sense.
So, for our example, \( -3 \times -4 \), we use this rule to determine the sign of the result.
Product Calculation
After understanding the absolute values and the sign rules, calculating the product is straightforward.
Here's how to proceed:
  • Identify the absolute values of the numbers: For \(-3(-4)\), we get \(|-3| = 3\) and \(|-4| = 4\)
  • Multiply the absolute values: \( 3 \times 4 = 12 \)
  • Apply the sign rule: Since both numbers were negative, the rule tells us our product will be positive.
So, the final product of \( -3 \) and \( -4 \) is \(+12\).
This step-by-step method ensures accuracy and helps solidify the understanding of multiplying negative numbers.

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Most popular questions from this chapter

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 6 if it is divisible by both 2 and \(3,\) and not otherwise. Show that (a) \(1,524,822\) is divisible by 6 and \((b) 2,873,590\) is not divisible by 6

The top of Mt. Whitney, visible from Death Valley, has an altitude of \(14,494\) ft above sea level. The bottom of Death Valley is 282 ft below sea level. Using 0 as sea level, find the difference between these two elevations.

Simplify each expression. $$ -10+x+4 x-7-4 x $$

Write each sentence as an equation, using \(x\) as the variable. Then find the solution from the set of integers between \(-12\) and \(12,\) inclusive. 6 less than a number is 4

Simplify each expression. $$ -5(8 x+2)-(5 x-3)-3 x+17 $$
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