Problem 41 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ (5.4)(-7.2) $$

Step-by-Step Solution

Answer

The product is -38.88.

Step 1: Identify the Numbers to be Multiplied

In this exercise, we need to multiply two real numbers: 5.4 and -7.2.

Step 2: Multiply the Absolute Values

First, ignore the negative sign and multiply the absolute values of the numbers: 5.4 and 7.2.\[5.4 \times 7.2 = 38.88\]

Step 3: Determine the Sign of the Answer

Since we are multiplying a positive number by a negative number, the result will be negative. Therefore, the product is -38.88.

Key Concepts

Multiplication of Real Numbers

Real numbers are numbers that include both rational and irrational numbers. When we perform multiplication with real numbers, we follow the same basic principles as with simple arithmetic.

Here's a quick rundown:

• Understand the numbers involved, paying attention to signs and absolute values as well.
• Multiply the absolute values of the numbers, regardless of their signs.
• Determine the sign of the product based on the rules for multiplication of signed numbers.
• A positive number multiplied by a negative number results in a negative product.
• A negative number multiplied by another negative number results in a positive product.
• Two positive numbers multiplied give a positive product.

Thus, multiplying real numbers is straightforward once you grasp the concepts of absolute value and sign alignment.

Absolute Value

Absolute value is the distance of a number from zero on the number line, regardless of direction. Absolute values are always positive or zero.

For example:

• The absolute value of 5.4 is 5.4 since it's already positive.
• The absolute value of -7.2 is 7.2 since absolute values make negative numbers positive.

When multiplying, calculating the absolute value is a beneficial step because it simplifies the numerical operation. You deal with positive numbers and worry about the sign afterward.

Negative Sign Multiplication

Dealing with negative numbers often complicates arithmetic operations. However, by understanding the rules, it becomes manageable.

Here are some key points:

• When multiplying numbers, if one of them is negative, the final product will also be negative.
• Conversely, if both numbers are negative, they cancel each other's negative effects, resulting in a positive answer.
• Always count the signs carefully to determine this.

In our example of (5.4)(-7.2), we already established the absolute values product is 38.88. But since one of those numbers was negative (the -7.2), the result will be negative, giving us a final product of -38.88.