Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$\frac{0}{-8}$$

Step-by-Step Solution

Answer

The result of \( \frac{0}{-8} \) is 0.

Step 1: Identify the expression

Firstly, identify the expression that we need to evaluate. Here, we are being asked to evaluate the expression \( \frac{0}{-8} \).

Step 2: Evaluate the numerator

The numerator of the expression is 0.

Step 3: Evaluate the denominator

The denominator of the expression is -8. In this case, the denominator is not 0, so the expression is defined and the division can be performed.

Step 4: Perform the division

Finally, divide the numerator by the denominator: \( \frac{0}{-8} = 0 \). Since 0 divided by any number is always 0, the answer is 0.

Key Concepts

Numerical Expressions

Numerical expressions are mathematical sentences involving only numbers and operations. These expressions can include addition, subtraction, multiplication, and division. Understanding numerical expressions is crucial because it is the foundation for solving more complex equations. Consider the expression \( \frac{0}{-8} \). Here, both the numerator and denominator are numbers: 0 and -8, respectively. No letters or variables are involved, which keeps it purely numerical.

To evaluate a numerical expression like this, identify each element involved. The key is to perform the operation indicated, in this case, division. First, check if the division can be performed, meaning the denominator is not zero. If it is not zero, like in this case, proceed by dividing. In \( \frac{0}{-8} \), 0 is divided by another number, resulting in 0. This highlights the rule that 0 divided by any non-zero number is always 0.

Division by Negative Numbers

Dividing by negative numbers can be confusing, but it follows the same rules as dividing by positive numbers, with the addition of managing signs. When you divide a positive number by a negative number, or vice versa, the result is negative. However, dividing zero by any negative number simplifies to zero, just as it does when dividing by a positive.

In the expression \( \frac{0}{-8} \), the negative number, -8, acts as the divisor. Since zero is being divided, the negative sign does not impact the outcome. Zero remains zero.

This outcome is consistent for any division where zero is the numerator, and it helps in understanding that zero is neutral and unaffected by the sign of the divisor.

Undefined Expressions

In mathematics, the term 'undefined' means the operation cannot produce a definitive result. An expression is often undefined if it involves division by zero. Division by zero is a critical concept to grasp because it raises a red flag in calculations.

If a denominator is zero, the expression cannot be calculated, because there is no number that can multiply with zero to return the numerator. For example, consider the expression \( \frac{x}{0} \). Regardless of what \(x\) is, the result remains undefined since dividing by zero disrupts the fundamental rules of arithmetic.

• Check for zero in the denominator before proceeding with division.


• Remember: division by zero is always undefined.

Fortunately, in \( \frac{0}{-8} \), the denominator is not zero. Thus, this expression is well-defined, and division can be safely performed, resulting in zero.