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### 6.5 Complex Fractions

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

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

CLASSROOM EXAMPLE 1 problem (Method 1).

Simplifying Complex Fractions (Method 1)

Solution:

Simplify each complex fraction.

Slide 6.5-6

CLASSROOM EXAMPLE 2 problem (Method 1).

Simplifying a Complex Fraction (Method 1)

Simplify the complex fraction.

Solution:

Slide 6.5-7

CLASSROOM EXAMPLE 3 problem (Method 1).

Simplifying a Complex Fraction (Method 1)

Simplify the complex fraction.

Solution:

Slide 6.5-8

Objective 2 problem (Method 1).

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

Slide 6.5-9

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 6.5-10

CLASSROOM EXAMPLE 4 denominator by the least common denominator (Method 2).

Simplifying Complex Fractions (Method 2)

Solution:

Simplify each complex fraction.

Slide 6.5-11

CLASSROOM EXAMPLE 5 denominator by the least common denominator (Method 2).

Simplifying a Complex Fraction (Method 2)

Simplify the complex fraction.

Solution:

Slide 6.5-12

CLASSROOM EXAMPLE 6 denominator by the least common denominator (Method 2).

Deciding on a Method and Simplifying Complex Fractions

Solution:

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 6.5-13

Objective 3 denominator by the least common denominator (Method 2).

Simplify rational expressions with negative exponents.

Slide 6.5-13

CLASSROOM EXAMPLE 7 denominator by the least common denominator (Method 2).

Simplifying Rational Expressions with Negative Exponents

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

Solution:

LCD = a2b3

Slide 6.5-14

CLASSROOM EXAMPLE 7 denominator by the least common denominator (Method 2).

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 6.5-15

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