Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$\frac{-21.6}{3}$$

Step-by-Step Solution

Answer

-7.2

Step 1: Identify the Numerator and Denominator

In \(\frac{-21.6}{3}\), -21.6 is the numerator and 3 is the denominator.

Step 2: Perform the division operation using the numerator and denominator

To find the result, divide -21.6 by 3, which equals -7.2.

Step 3: Write the Final Answer

So, \(\frac{-21.6}{3}\) equals -7.2

Key Concepts

Undefined Expressions

When performing arithmetic operations like division, we may sometimes encounter what is known as an undefined expression. An expression is undefined when it does not result in a real number or when the operation cannot be performed according to mathematical rules.

In division, the most common undefined expression is the division by zero. For example, if you try to divide any number by zero, like \(\frac{5}{0}\), you won't get a real number as a result. This is because dividing by zero has no meaning in the realm of real numbers—it's like asking, 'how many times does zero go into a number?' The answer is indefinite, and hence, the operation is undefined.

To ensure a student avoids encountering undefined expressions, always remember that the denominator – the number you are dividing by – should never be zero.

Numerator and Denominator

The division of two numbers can be written as a fraction where the number on top is called the numerator and the number at the bottom is the denominator. Understanding these terms is crucial as they are the key components in any division operation.

In the given exercise, the number -21.6 is the numerator, which represents the quantity to be divided, while 3 is the denominator that signifies the number of parts the numerator is to be divided into. It is important to recognize these parts as it impacts how the division is approached. For instance, if their roles were reversed, the result would be quite different.

Visualizing Division

Visual aids can help to grasp this concept further. Imagine you have -21.6 apples that need to be distributed evenly into 3 baskets. The problem then asks how many apples would go into each basket. This illustrates the essence of the numerator and the denominator in a practical way.

Division Operation

The division operation is essentially a method of splitting a number (the numerator) into equal parts (as designated by the denominator). The division is one of the basic operations of arithmetic and can also be considered the inverse operation of multiplication.

Going back to our problem, executing the division operation involves taking the numerator, -21.6, and dividing it by the denominator, 3. This can be done using long division or, as in most modern scenarios, a calculator. The key is to determine how many times 3 can be multiplied to reach -21.6. The result of this operation is -7.2, which means that -7.2 is the value of each part when -21.6 is divided into 3 equal parts.

Checking Your Work

A healthy habit is to multiply the result by the denominator to see if it returns you to the numerator. In this case, -7.2 times 3 indeed gives us -21.6, signifying the operation was performed correctly.