Problem 35 - Intermediate Algebra

Question

Perform the operations. See Example 3 . $$ -5(6)(-2) $$

Step-by-Step Solution

Answer

The result of the multiplication is 60.

Step 1: Understand the Problem

The problem involves multiplying three numbers: 1. \(-5\) 2. \(6\) 3. \(-2\). The task is to perform these operations correctly.

Step 2: Multiply the First Two Numbers

First, multiply the first two numbers, \(-5\) and \(6\): \[-5 imes 6 = -30\]This result comes from multiplying a negative number by a positive number, which produces a negative result.

Step 3: Multiply the Result with the Third Number

Next, take the result from Step 2, \(-30\), and multiply it by the third number, \(-2\): \[-30 imes -2 = 60\]Multiplying two negative numbers results in a positive number.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They have a minus sign (−) in front of them. Understanding how negative numbers work is key to solving arithmetic problems involving multiplication and other operations. Here's what you should know about negative numbers:

• They represent values below zero, often used in real life to indicate debt or temperature below freezing point.
• When multiplying two negative numbers, the result is always a positive number.
• Multiplying a negative number by a positive number results in a negative number.

For instance, if you multiply \[-5 \times -2\], the result is \[10\], since two negative numbers make a positive.
On the other hand, multiplying negative \[-5\] by positive \[6\] gives \[-30\], because a negative times a positive is negative.

Positive Integers

Positive integers are numbers greater than zero and are usually used to count or order objects. They do not have any sign in front, but are sometimes denoted by a positive sign (+). Here's how positive integers feature in multiplication:

• They are whole numbers starting from 1 upwards, such as 1, 2, 3, etc.
• The product of two or more positive integers is always positive.
• In arithmetic operations, they contribute directly to the size of the final result.

For example, multiplying the positive integer \[6\] with the negative number \[-5\] in the problem leads to \[-30\].
Here, \[6\] is the factor that scales the size of \[-5\] but keeps the negative sign due to the initial factor being negative.

Arithmetic Operations

Arithmetic operations include basic mathematical procedures such as addition, subtraction, multiplication, and division. For multiplication, understanding how signs interact helps in determining the result:

• The product of two numbers with the same sign (positive or negative) is positive.
• The product of two numbers with different signs is negative.
• Order of multiplication doesn’t affect the final result (commutative property).

In our exercise, multiply \[-5\] by \[6\] to get \[-30\] and then multiply by another negative \[-2\], resulting in a positive \[60\].
This process shows how using arithmetic rules allows us to handle multiple products logically and systematically.