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6.5 Complex Fractions

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**Complex Fractions. Complex Fraction**

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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**Method 1 for Simplifying a Complex Fraction**

Simplify a complex fraction by writing it as a division problem (Method 1). Since the main fraction bar represents division in a complex fraction, one method of simplifying a complex fraction involves division. 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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**Simplifying Complex Fractions (Method 1)**

CLASSROOM EXAMPLE 1 Simplifying Complex Fractions (Method 1) Simplify each complex fraction. Solution: Slide 6.5-6

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**Simplifying a Complex Fraction (Method 1)**

CLASSROOM EXAMPLE 2 Simplifying a Complex Fraction (Method 1) Simplify the complex fraction. Solution: Slide 6.5-7

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**Simplifying a Complex Fraction (Method 1)**

CLASSROOM EXAMPLE 3 Simplifying a Complex Fraction (Method 1) Simplify the complex fraction. Solution: Slide 6.5-8

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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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**Method 2 for Simplifying a Complex Fraction**

Simplify a complex fraction by multiplying numerator and denominator by the least common denominator (Method 2). 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. 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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**Simplifying Complex Fractions (Method 2)**

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

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**Simplifying a Complex Fraction (Method 2)**

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

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**Simplify each complex fraction.**

CLASSROOM EXAMPLE 6 Deciding on a Method and Simplifying Complex Fractions Simplify each complex fraction. Solution: 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

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**Simplify rational expressions with negative exponents.**

Objective 3 Simplify rational expressions with negative exponents. Slide

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**Simplify the expression, using only positive exponents in the answer.**

CLASSROOM EXAMPLE 7 Simplifying Rational Expressions with Negative Exponents Simplify the expression, using only positive exponents in the answer. Solution: LCD = a2b3 Slide

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**Simplify the expression, using only positive exponents in the answer.**

CLASSROOM EXAMPLE 7 Simplifying Rational Expressions with Negative Exponents (cont’d) Simplify the expression, using only positive exponents in the answer. Solution: Write with positive exponents. LCD = x3y Slide

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