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6.5 Complex Fractions

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Complex Fractions. The quotient of two mixed numbers in arithmetic, such as can be written as a fraction. In algebra, some rational expressions also have fractions in the numerator, or denominator, or both. Complex Fraction A quotient with one or more fractions in the numerator, or denominator, or both is called a complex fraction. The parts of a complex fraction are named as follows. Numerator of complex fraction Main fraction bar Denominator of complex fraction Slide 6.5-3

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Objective 1 Simplify a complex fraction by writing it as a division problem (Method 1). Slide 6.5-4

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Since the main fraction bar represents division in a complex fraction, one method of simplifying a complex fraction involves division. Simplify a complex fraction by writing it as a division problem (Method 1). Method 1 for Simplifying a Complex Fraction Step 1: Write both the numerator and denominator as single fractions. Step 2: Change the complex fraction to a division problem. Step 3: Perform the indicated division. Slide 6.5-5

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Simplify each complex fraction. Solution : Slide Simplifying Complex Fractions (Method 1) CLASSROOM EXAMPLE 1

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Simplify the complex fraction. Solution: Slide Simplifying a Complex Fraction (Method 1) CLASSROOM EXAMPLE 2

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Solution : Simplify the complex fraction. Slide Simplifying a Complex Fraction (Method 1) CLASSROOM EXAMPLE 3

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Objective 2 Simplify a complex fraction by multiplying numerator and denominator by the least common denominator (Method 2). Slide 6.5-9

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Since any expression can be multiplied by a form of 1 to get an equivalent expression, we can multiply both the numerator and denominator of a complex fraction by the same nonzero expression to get an equivalent rational expression. If we choose the expression to be the LCD of all the fractions within the complex fraction, the complex fraction will be simplified. Simplify a complex fraction by multiplying numerator and denominator by the least common denominator (Method 2). Method 2 for Simplifying a Complex Fraction Step 1: Find the LCD of all fractions within the complex fraction. Step 2: Multiply both the numerator and denominator of the complex fraction by this LCD using the distributive property as necessary. Write in lowest terms. Slide

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Solution: Simplify each complex fraction. Slide Simplifying Complex Fractions (Method 2) CLASSROOM EXAMPLE 4

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Simplify the complex fraction. Solution: Slide Simplifying a Complex Fraction (Method 2) CLASSROOM EXAMPLE 5

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Simplify each complex fraction. Remember the same answer is obtained regardless of whether Method 1 or Method 2 is used. Some students prefer one method over the other. Slide Deciding on a Method and Simplifying Complex Fractions Solution: CLASSROOM EXAMPLE 6

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Simplify rational expressions with negative exponents. Objective 3 Slide

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Simplify the expression, using only positive exponents in the answer. LCD = a 2 b 3 Slide CLASSROOM EXAMPLE 7 Simplifying Rational Expressions with Negative Exponents Solution:

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Write with positive exponents. LCD = x 3 y Slide CLASSROOM EXAMPLE 7 Simplifying Rational Expressions with Negative Exponents (contd) Simplify the expression, using only positive exponents in the answer. Solution:

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