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Chapter 4: Problem 7

Reduce each of the following rational expressions to lowest terms. $$\frac{-9}{21}$$

Short Answer

Expert verified
The reduced form is \(\frac{-3}{7}\).

Step by step solution

01

Identify the Numerator and Denominator

The given rational expression is \(\frac{-9}{21}\). Identify the numerator, which is -9, and the denominator, which is 21.
02

Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor (GCD) of the numerator and denominator. The GCD of 9 and 21 is 3.
03

Divide Both Terms by the GCD

Divide the numerator and the denominator by their GCD. \(\frac{-9 \div 3}{21 \div 3} = \frac{-3}{7}\)
04

Simplify the Expression

After dividing, simplify the expression to \(\frac{-3}{7}\). This is the rational expression in its lowest terms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
When simplifying rational expressions, finding the greatest common divisor (GCD) of the numerator and the denominator is crucial. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder. For example, with the numbers 9 and 21, the GCD is 3. Identifying the GCD allows you to simplify the fraction to its lowest terms, making the expression easier to work with.

Here's how to find the GCD:
  • List the factors of each number.
    E.g., Factors of 9: 1, 3, 9
    Factors of 21: 1, 3, 7, 21
  • Select the largest common factor, which is 3 in this case.
Finding the GCD is a key step in fraction simplification because it ensures that you reduce the fraction to its simplest form.
Numerator and Denominator
Understanding the roles of the numerator and the denominator is vital in working with fractions and rational expressions. In any given fraction, the numerator is the number on the top, and the denominator is the number on the bottom. For the rational expression \(\frac{-9}{21}\), -9 is the numerator, and 21 is the denominator.

Here are some important points to remember:
  • The numerator represents how many parts of the whole are taken.
  • The denominator indicates the total number of equal parts the whole is divided into.
  • Both terms are crucial in simplifying fractions, as you generally reduce both by their GCD.
Knowing which value is the numerator and which is the denominator helps in applying mathematical operations correctly.
Fraction Simplification
Simplifying a fraction, also known as reducing it to its lowest terms, makes the fraction easier to work with. The process involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, given the fraction \(\frac{-9}{21}\), the GCD is 3.

Here are the steps to simplify the fraction:
  • Identify the numerator and the denominator.
  • Find the GCD of both numbers.
  • Divide the numerator and the denominator by the GCD.
    For \(\frac{-9}{21}\), dividing by 3 gives \(\frac{-3}{7}\).
The simplified version of \(\frac{-9}{21}\) is \(\frac{-3}{7}\). Simplification reduces complexity and makes subsequent mathematical operations more straightforward.

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Most popular questions from this chapter

Solve each of the following problems algebraically. Be sure to label what the variable represents. The length of a rectangle is \(5 \mathrm{cm}\) more than twice the width. If the perimeter of the rectangle is \(34 \mathrm{cm},\) find its dimensions.

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. A car that is insured by the Reliable Auto Insurance Company is damaged in an accident. The company will pay for damages to the car up to \(75 \%\) of its Kelley Blue Book value. If the company pays \(\$ 8700\) on a certain claim, what can you say about the Kelley Blue Book value of the car?

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Two trains leave two cities that are \(400 \mathrm{km}\) apart. They both leave at 11: 00 A.M. traveling toward each other on parallel tracks. If one train travels at \(70 \mathrm{kph}\) and the other travels at \(90 \mathrm{kph}\), at what time will they pass each other?

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{7}{6 a^{2} b}+\frac{3}{4 a b^{3}}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$5 y^{3}-\frac{2}{y}+\frac{1}{2 y}$$
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