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Chapter 7 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Least Common Denominators Find the least common denominator for a group of fractions. Rewrite rational expressions with given denominators. 1 1 2 27.37.3

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Find the least common denominator for a group of fractions. Slide 7.3 - 3

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the least common denominator for a group of fractions. Adding or subtracting rational expressions often requires a least common denominator (LCD), the simplest expression that is divisible by all of the denominators in all of the expressions. For example, the least common denominator for the fractions and is 36, because 36 is the smallest positive number divisible by both 9 and 12. Slide 7.3 - 4 We can often find least common denominators by inspection. For example, the LCD for and is 6m. In other cases, we find the LCD by a procedure similar to that used in Section 6.1 for finding the greatest common factor.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the least common denominator for a group of fractions. (cont’d) To find the least common denominator, use the following steps. Step 1: Factor each denominator into prime factors. Slide 7.3 - 5 Step 2: List each different denominator factor the greatest number of times it appears in any of the denominators. Step 3: Multiply the denominator factors from Step 2 to get the LCD. When each denominator is factored into prime factors, every prime factor must be a factor of the least common denominator.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Find the LCD for each pair of fractions. Solution: Finding the LCD Slide 7.3 - 6

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the LCD for EXAMPLE 2 Finding the LCD Slide 7.3 - 7 Solution: When finding the LCD, use each factor the greatest number of times it appears in any single denominator, not the total number of times it appears.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Finding the LCD Slide 7.3 - 8 Find the LCD for the fractions in each list. Either x − 1 or 1 − x, since they are opposite expressions.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Rewrite rational expressions with given denominators. Slide 7.3 - 9

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rewrite rational expressions with given denominators. Once the LCD has been found, the next step in preparing to add or subtract two rational expressions is to use the fundamental property to write equivalent rational expressions. Step 1: Factor both denominators. Slide 7.3 - 10 Step 2: Decide what factor(s) the denominator must be multiplied by in order to equal the specified denominator. Step 3: Multiply the rational expression by the factor divided by itself. (That is, multiply by 1.)

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rewrite each rational expression with the indicated denominator. EXAMPLE 4 Solution: Writing Rational Expressions with Given Denominotors Slide 7.3 - 11

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Rewrite each rational expression with the indicated denominator. Solution: Writing Rational Expressions with Given Denominators Slide 7.3 - 12

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