Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$-0.2(-0.6)$$

Step-by-Step Solution

Answer

The product of -0.2 and -0.6 is \(0.12\).

Step 1: Identify the Numbers

Identify the two numbers to be multiplied, which are -0.2 and -0.6.

Step 2: Multiply the absolute values

Multiply the absolute values of the two numbers: 0.2 * 0.6 = 0.12.

Step 3: Determine the sign

Since both numbers are negative, their product is positive as per the rule in mathematics, which states the multiplication of two negative numbers gives a positive number.

Key Concepts

Understanding Absolute Values

When dealing with negative decimals, the first step in multiplication is to focus on their absolute values.
Absolute value refers to the distance of a number from zero on the number line, regardless of its direction (positive or negative).
This means that the absolute value is always a positive number or zero if the number itself is zero.
To find the absolute value of a negative decimal like \(-0.2\), you simply remove the negative sign, making it \(0.2\). Similarly, the absolute value of \(-0.6\) is \(0.6\).
By considering absolute values, we temporarily set aside the negative signs to simplify calculations.
This aspect significantly reduces any confusion during multiplication.
Recognizing and understanding absolute values is an essential mathematical skill that

1. simplifies complex problems, and
2. helps focus solely on magnitude.

Mastering the Rules of Multiplication

The rules of multiplication are foundational principles that guide the process of multiplying two numbers.
When multiplying numbers, regardless of their type—integers, decimals, or fractions—always begin by multiplying their absolute values.
For our example, - Multiply the numbers \(0.2\) and \(0.6\). - Calculate the product: \(0.2 \times 0.6 = 0.12\).
These rules are consistent, whether you are dealing with whole numbers or decimals.
By maintaining consistency, you ensure that your calculations remain accurate.
The beauty of mathematics is in its universality, and these principles hold true in every scenario:

• The absolute values get multiplied first,
• order of multiplication does not affect the result, and
• precision in calculation is key.

Decoding Sign Determination

After calculating the product of the absolute values, it becomes important to determine the sign of the final answer.
This step determines whether your result will be positive or negative.
In multiplication rules, the sign of the product is determined by the signs of the numbers being multiplied.
Here are the basic rules regarding sign determination:

• If both numbers are positive, the product is positive.
• If one number is positive and the other is negative, the product is negative.
• If both numbers are negative, like in this example, the product is positive.

Applying these rules to our problem, since both \(-0.2\) and \(-0.6\) are negative, their multiplication results in a positive product of \(0.12\).
This simple method of rule application ensures that even with various combinations of positive and negative numbers, finding the correct sign becomes effortless.