Copyright © Cengage Learning. All rights reserved. Rational Expressions and Equations; Ratio and Proportion 6

Copyright © Cengage Learning. All rights reserved. Section 6.4 Simplifying Complex Fractions

3 Objectives Simplify a complex fraction. Simplify a fraction containing terms with negative exponents

4 Simplify a complex fraction 1.

5 Simplify a complex fraction Fractions such as that contain fractions in their numerators and/or denominators are called complex fractions. Complex fractions should be simplified.

6 Simplify a complex fraction For example, we can simplify by doing the division: There are two methods that we can use to simplify complex fractions.

7 Simplify a complex fraction Simplifying Complex Fractions Method 1 Write the numerator and the denominator of the complex fraction as single fractions. Then divide the fractions and simplify. Method 2 Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in its numerator and denominator. Then simplify the results, if possible.

8 Simplify a complex fraction Using Method 1 to simplify (assuming no division by 0), we proceed as follows: Write 1 as and 2 as. Add the fractions in the numerator and subtract the fractions in the denominator.

9 Simplify a complex fraction Invert the divisor and multiply. Multiply the fractions. Write the complex fraction as an equivalent division problem. Divide out the common factor of 5: = 1.

10 Simplify a complex fraction To use Method 2, we first determine that the LCD of the fractions in the numerator and denominator is 5. We then multiply both the numerator and denominator by 5. Multiply both numerator and denominator by 5. Simplify. Use the distributive property to remove parentheses.

11 Example Simplify:. Assume that no denominators are 0. Solution: We will simplify the complex fraction using both methods. Method 1

12 Example – Solution Method 2 cont’d

13 Simplify a fraction containing terms with negative exponents 2.

14 Simplify a fraction containing terms with negative exponents Many fractions with terms containing negative exponents are complex fractions as the next example illustrates.

15 Example Simplify:. Assume no denominator is 0. Solution: We will write each expression using positive exponents and then simplify the complex fraction using Method 2: Write without negative exponents.

16 Example – Solution The result cannot be simplified. Therefore either of the last two steps is a correct answer. Distribute. Factor the numerator and denominator. Multiply numerator and denominator of the complex fraction by x 2 y 2, the LCD. cont’d