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Chapter 6: Problem 20

Reduce each rational expression to its lowest terms. $$\frac{14}{91}$$

Short Answer

Expert verified
The reduced form is \( \frac{2}{13} \).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

First, identify the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 14 and 91 is 7.
02

Divide the Numerator by the GCD

Divide the numerator (14) by the GCD (7). \[ \frac{14}{7} = 2 \]
03

Divide the Denominator by the GCD

Divide the denominator (91) by the GCD (7). \[ \frac{91}{7} = 13 \]
04

Write the Reduced Form

Combine the results of the divisions to write the rational expression in its reduced form. \[ \frac{14}{91} = \frac{2}{13} \]

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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 crucial concept in simplifying rational expressions. It refers to the largest number that can exactly divide both the numerator and the denominator without leaving a remainder. Finding the GCD can greatly simplify fractions. For example, in the fraction \( \frac{14}{91} \), the GCD of 14 and 91 is determined to be 7. Identifying the GCD correctly allows us to reduce the fraction more efficiently.
Here are steps to find the GCD:
  • List all factors of both the numerator and the denominator.
  • Identify the largest common factor.
This process helps in simplifying fractions quickly and accurately.
numerator
The numerator is the top part of a fraction. It tells how many parts we have out of a whole. In the fraction \( \frac{14}{91} \), 14 is the numerator.
When simplifying fractions, we first focus on the numerator to check for common factors with the denominator. For instance, to simplify \( \frac{14}{91} \), we divide the numerator (14) by the GCD, which is 7:
\[ \frac{14}{7} = 2 \] This gives us a simplified numerator. Always remember to include this step in fraction simplification.
denominator
The denominator is the bottom part of a fraction. It shows into how many parts the whole is divided. In the fraction \( \frac{14}{91} \), 91 is the denominator.
To reduce a fraction, we must simplify the denominator alongside the numerator. Using our example, we divide the denominator (91) by the GCD, which is 7:
\[ \frac{91}{7} = 13 \] This results in a simplified denominator. Simplifying both numerator and denominator is key to reducing fractions correctly.
Fraction Simplification
Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common divisors other than 1. To simplify a fraction, follow these steps:
  • Find the GCD of the numerator and denominator.
  • Divide both the numerator and the denominator by the GCD.
Using our example \( \frac{14}{91} \):
First, find the GCD of 14 and 91, which is 7.
Then, divide both the numerator and the denominator by 7:
\[ \frac{14 \div 7}{91 \div 7} = \frac{2}{13} \] The fraction \( \frac{14}{91} \) simplifies to \( \frac{2}{13} \). Breaking down each step ensures accurate and clear simplification.

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Most popular questions from this chapter

Discussion. For each equation, find the values for \(x\) that cannot be solutions to the equation. Do not solve the equations. a) \(\frac{1}{x}+\frac{1}{x-1}=\frac{1}{2}\) b) \(\frac{x}{x-1}=\frac{1}{2}\) c) \(\frac{1}{x^{2}+1}=\frac{1}{x+1}\)

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{3 a+1}}{y^{2 b-3}} \cdot \frac{y^{3 b+4}}{x^{2 a-1}}$$

Solve each equation. $$\frac{-3}{2 x}=\frac{1}{-5}$$

Which of the following equations is not an identity? Explain. a) \(\frac{x^{2}-1}{2} \cdot \frac{2}{x-1}=x+1\) b) \(\frac{x-1}{x^{2}-1}=x+1\) c) \(x^{2}-1=(x-1)(x+1)\) d) \(\frac{1}{x^{2}-1} \div \frac{1}{x+1}=\frac{1}{x-1}\)

Perform the indicated operations. $$\frac{x^{2}+5 x+6}{x} \cdot \frac{x^{2}}{3 x+6} \cdot \frac{9}{x^{2}-4}$$
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