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Chapter 6: Problem 54

Simplifying a complex fraction by Method 1 is a good way to review the methods of adding, subtracting, multiplying, and dividing rational expressions. Method 2 gives a good review of the fundamental property of rational numbers. $$ \frac{\frac{4}{m}+\frac{m+2}{m-1}}{\frac{m+2}{m}-\frac{2}{m-1}} $$ The answers for Exercises 51 and 53 should be the same. Which method do you prefer? Explain why.

Short Answer

Expert verified
\( \frac{m^2 + 6m - 4}{m^2 - m - 2} \)

Step by step solution

01

Identify the complex fraction

The given complex fraction is \[ \frac{\frac{4}{m}+\frac{m+2}{m-1}}{\frac{m+2}{m}-\frac{2}{m-1}} \]
02

Find a common denominator for the numerator

To combine \( \frac{4}{m} \) and \( \frac{m+2}{m-1} \), find the common denominator. The common denominator is \( m(m-1) \). Rewrite the fractions: \[ \frac{4(m-1) + m(m+2)}{m(m-1)} \]
03

Simplify the numerator

Combine the terms in the numerator: \[ \frac{4(m-1) + m(m+2)}{m(m-1)} = \frac{4m-4 + m^2 + 2m}{m(m-1)} = \frac{m^2 + 6m - 4}{m(m-1)} \]
04

Find a common denominator for the denominator

To combine \( \frac{m+2}{m} \) and \( \frac{2}{m-1} \), find the common denominator. The common denominator is \( m(m-1) \). Rewrite the fractions: \[ \frac{(m+2)(m-1) - 2m}{m(m-1)} \]
05

Simplify the denominator

Combine the terms in the denominator: \[ \frac{(m+2)(m-1) - 2m}{m(m-1)} = \frac{m^2 - m + 2m - 2 - 2m}{m(m-1)} = \frac{m^2 - m - 2}{m(m-1)} \]
06

Simplify the complex fraction

Divide the simplified numerator by the simplified denominator:\[ \frac{\frac{m^2 + 6m - 4}{m(m-1)}}{\frac{m^2 - m - 2}{m(m-1)}} = \frac{m^2 + 6m - 4}{m^2 - m - 2} \]
07

Final simplification

Recognize that \( m^2 + 6m - 4 \) and \( m^2 - m - 2 \) do not factor equally, so the simplified form is indeed: \[ \frac{m^2 + 6m - 4}{m^2 - m - 2} \]

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Key Concepts

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

Adding Rational Expressions
Adding rational expressions involves finding a common denominator. This is crucial because you can only add fractions when they share the same denominator. For example, when adding \(\frac{4}{m}\) and \(\frac{m+2}{m-1}\), we need to identify their least common denominator (LCD). The LCD for these fractions is \m(m-1).\ To add them:
1. Rewrite each fraction with the common denominator:
  • \frac{4(m-1)}{m(m-1)}\
  • \frac{m(m+2)}{m(m-1)}\
2. Combine numerators over the common denominator:
  • \frac{4(m-1) + m(m+2)}{m(m-1)}\ = \frac{4m-4 + m^2 + 2m}{m(m-1)}\
3. Simplify the result:
  • \frac{m^2 + 6m - 4}{m(m-1)}\

This method helps you practice the key skill of manipulating and simplifying fractions.
Subtracting Rational Expressions
Subtracting rational expressions is similar to adding them. You also need a common denominator here. Taking the fractions \(\frac{m+2}{m}\) and \(\frac{2}{m-1}\) as an example:
1. Identify the LCD for these fractions, which in this case is \m(m-1).\
2. Rewrite the fractions with the common denominator:
  • \frac{(m+2)(m-1)}{m(m-1)}\
  • \frac{2m}{m(m-1)}\
3. Subtract the numerators over the common denominator:
  • \frac{(m+2)(m-1) - 2m}{m(m-1)}\
4. Simplify the result:
  • \frac{m^2 - m + 2m - 2 - 2m}{m(m-1)} = \frac{m^2 - m - 2}{m(m-1)}\
This approach helps you become proficient in dealing with rational expressions and preparing for more advanced algebraic operations.
Multiplying Rational Expressions
Multiplying rational expressions is relatively straightforward. Multiply the numerators together and the denominators together. For example, multiplying \(\frac{a}{b}\) and \(\frac{c}{d}\):
1. Multiply the numerators:
  • \a \cdot c\
2. Multiply the denominators:
  • \b \cdot d\
3. Combine them into a single fraction:
  • \frac{a \cdot c}{b \cdot d}\
If we were to multiply more complex expressions, the process remains the same. Just remember to simplify the resulting fraction if possible. Factoring beforehand can simplify the multiplication.
Dividing Rational Expressions
Dividing rational expressions involves multiplying by the reciprocal. For instance, to divide \(\frac{a}{b} \div \frac{c}{d}\), you would:
1. Rewrite the division as multiplication by the reciprocal:
  • \frac{a}{b} \times \frac{d}{c}\
2. Multiply the fractions:
  • Numerator: \( a \cdot d \)
  • Denominator: \( b \cdot c \)
3. Combine and simplify:
  • \frac{a \cdot d}{b \cdot c}\
Practicing division of rational expressions enhances your understanding and gets you comfortable with inverting and multiplying complex fractions.
Common Denominators
A common denominator is essential for adding or subtracting rational expressions. Given fractions with different denominators, finding the LCD allows you to combine them. Here's how you can find a common denominator:
  • Identify the denominators of the fractions you are working with.
  • Determine the least common multiple (LCM) of those denominators. This will be your LCD.
  • Rewrite each fraction with the LCD.
For instance, for \(\frac{4}{m}\) and \(\frac{m+2}{m-1}\), the LCD is \m(m-1).\ Using the LCD:
  • Rewrite each fraction: \frac{4(m-1)}{m(m-1)}\ and \frac{m(m+2)}{m(m-1)}\
  • Now they can be easily added together as their denominators match.
Mastering common denominators lays a strong foundation for handling any rational expression operations you will encounter.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{(x+4)(x-3)}{(x+5)(x+4)} $$

Write each rational expression in lowest terms. $$ \frac{(8-p)(x+2)}{(p-8)(x-2)} $$

Write each rational expression in lowest terms. $$ \frac{y^{2}-5 y-14}{y^{2}+y-2} $$

Multiply or divide as indicated. $$ \frac{(6 x+5)(x-3)}{(x-1)(x-3)} \div \frac{(2 x+7)(x+9)}{(x-1)(x+9)} $$

Which rational expressions are equivalent to \(-\frac{x}{y} ?\) A. \(\frac{-x}{-y}\) B. \(\frac{x}{-y}\) C. \(\frac{x}{y}\) D. \(-\frac{x}{-y}\) E. \(\frac{-x}{y}\) F. \(-\frac{-x}{-y}\)
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