Problem 53 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$\frac{0}{-7}$$

Step-by-Step Solution

Answer

The answer is 0.

Step 1: Identify the numerator and the denominator

In this division problem, the numerator is 0 and the denominator is -7.

Step 2: Verify if the division is undefined

The division would be undefined only if the denominator was 0. Here, the denominator is -7, so the division is defined.

Step 3: Perform the division

For any number \(a\), it is known that \(0/a = 0\). This is because zero divided by any non-zero number always equals zero. Thus, \(0/(-7) = 0\).

Key Concepts

Numerator and Denominator

In algebraic division, understanding the roles of the numerator and the denominator is crucial. The numerator is the number or expression on top of the fraction bar. It represents the number being divided. In our example, \( \frac{0}{-7} \), the numerator is 0.The denominator is the number or expression below the fraction bar. It shows how many parts the numerator is divided into. In the given fraction, the denominator is -7. This separates our fraction into -7 parts, theoretically speaking. Knowing which number is the numerator and which is the denominator helps identify proper division scenarios and consequences when dividing numbers.

Defined and Undefined Expressions

Not all algebraic divisions result in a valid number. This happens because certain expressions can be undefined. An expression becomes undefined if it divides a number by zero.For instance, consider the fraction \( \frac{a}{b} \). This expression is defined if \( b eq 0 \). In cases where \( b = 0 \), the expression is undefined because you cannot divide by zero.In the original exercise, \( \frac{0}{-7} \) is a defined expression because the denominator is -7, not 0. Whenever analyzing expressions, always check the denominator to confirm it's non-zero for the division to be valid.

Zero Division Rule

One key rule in division is the zero division rule. This rule states that zero divided by any non-zero number is always zero. Consider a case \( 0/a \), where \( a eq 0 \). The result is simply 0.In our problem \( \frac{0}{-7} \), according to the zero division rule, dividing 0 by -7 gives a result of 0. This is because dividing zero into any number of parts still gives you zero. Remembering the zero division rule provides a quick way to solve division problems, saving time and avoiding errors.