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Chapter 4: Problem 10

Reduce each of the following rational expressions to lowest terms. $$\frac{-14}{-8}$$

Short Answer

Expert verified
The reduced form of the rational expression is \(\frac{7}{4}\).

Step by step solution

01

Identify the Rational Expression

The given rational expression is \(\frac{-14}{-8}\). First, observe that both the numerator and the denominator have a negative sign.
02

Simplify the Signs

Since both the numerator and the denominator are negative, the expression can be simplified by cancelling out the negative signs: \(\frac{-14}{-8} = \frac{14}{8}\).
03

Find the Greatest Common Divisor (GCD)

Determine the GCD of the numerator (14) and the denominator (8). The factors of 14 are 1, 2, 7, and 14. The factors of 8 are 1, 2, 4, and 8. The common greatest factor is 2.
04

Divide by the GCD

Divide both the numerator and denominator by their GCD (2): \(\frac{14}{8} = \frac{14 \div 2}{8 \div 2} = \frac{7}{4}\).
05

Write the Reduced Expression

The rational expression in its lowest terms is \(\frac{7}{4}\).

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Key Concepts

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

Greatest Common Divisor
The Greatest Common Divisor (GCD) is a key concept in simplifying rational expressions. It refers to the largest number that can evenly divide both the numerator and the denominator of a fraction.
For example, in the rational expression \(\frac{14}{8}\), we need to find the GCD of 14 and 8.
First, list the factors of each number:
  • Factors of 14: 1, 2, 7, 14
  • Factors of 8: 1, 2, 4, 8
The greatest common factor between these two sets is 2.
Hence, 2 is the GCD. Dividing both the numerator and the denominator by their GCD helps in reducing the fraction to its simplest form.
Rational Expressions
A rational expression is a fraction that consists of polynomials in the numerator and the denominator.
It is of the form \(\frac{P(x)}{Q(x)}\), where \P(x)\ and \Q(x)\ are polynomials.
In our example, \(\frac{-14}{-8}\), both the numerator and the denominator are integer values, which are simply polynomials of degree 0.
Rational expressions can often be simplified by eliminating common factors in the numerator and denominator.
This process generally involves factoring polynomials and finding the GCD as outlined above.
Fraction Simplification
Simplifying a fraction makes it easier to understand and use. To simplify a fraction:
  • Identify and cancel common factors in both the numerator and denominator.
  • Divide both parts of the fraction by their GCD.
Let's revisit our exercise \(\frac{-14}{-8}\).
First, cancel out the negative signs: \(\frac{-14}{-8} = \frac{14}{8}\).
Then, find the GCD of 14 and 8, which is 2.
Finally, divide both numerator and denominator by 2: \(\frac{14}{8} = \frac{14 \div 2}{8 \div 2} = \frac{7}{4}\).
The simplified expression is \(\frac{7}{4}\). Simplifying fractions helps in making calculations easier and the results more interpretable.

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Most popular questions from this chapter

Solve the equations and inequalities. $$3-\frac{a+1}{4}=\frac{a+4}{2}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{5}{4 a b^{2}}+\frac{9}{10 a^{3} b}$$

Solve the equations and inequalities. $$0.6(x-4)-0.4(x-5)=4.6$$

Solve the equations and inequalities. $$\frac{t}{6}+\frac{t+3}{5}+\frac{t-2}{10}=t-6$$

If the exercise is an equation, solve it; if not, perform the indicated operations and express your answer as a single fraction. $$\frac{x+5}{2}-\frac{x-1}{4}=2$$
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