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5.3: Simplify Complex Rational Expressions

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Learning Objectives

By the end of this section, you will be able to:

  • Simplify a complex rational expression by writing it as division
  • Simplify a complex rational expression by using the LCD
Be Prepared

Before you get started, take this readiness quiz.

  1. Simplify: 35910.
  2. Simplify: 11342+4·5.
  3. Solve: 12x+14=18.

Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:

3458 and x2xy6.

In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

Definition 5.3.1

A complex rational expression is a rational expression in which the numerator and/or the denominator contains a rational expression.

Here are some complex rational expressions:

4y38y29,1x+1yxyyx and 2x+64x64x236.

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

We have already seen this complex rational expression earlier in this chapter:

6x27x+24x82x28x+3x25x+6.

We noted that fraction bars tell us to divide, and rewrote it as the division problem:

(6x27x+24x8)÷(2x28x+3x25x+6).

Then, we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.

Example 5.3.2

Simplify the complex rational expression by writing it as division: 6x43x216.

Solution
  6x43x216
Rewrite the complex fraction as division. =6x4÷3x216
Rewrite as the product of first times the reciprocal of the second.

=6x4x2163

Factor.

=32x4(x4)(x+4)3

Multiply. =32(x4)(x+4)3(x4)
Remove common factors.

=32(x4)(x+4)3(x4)

Simplify.

=2(x+4)

Are there any value(s) of x that should not be allowed? The original complex rational expression had denominators of x4 and x216 This expression would be undefined if x=4 or x=4.

Try It 5.3.3

Simplify the complex rational expression by writing it as division: 2x213x+1.

Answer

23(x1)

Try It 5.3.4

Simplify the complex rational expression by writing it as division: 1x27x+122x4.

Answer

12(x3)

Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

Example 5.3.5

Simplify the complex rational expression by writing it as division: 13+161213.

Solution
  13+161213

Find the LCD and add the fractions in the numerator. Find the LCD and subtract the fractions in the denominator.

=1232+1613231232
Simplify the numerator and denominator.

=26+163626

=3616

Rewrite the complex rational expression as a division problem. =36÷16
Multiply the first by the reciprocal of the second. =3661
Simplify. =3
Try It 5.3.6

Simplify the complex rational expression by writing it as division: 12+2356+112

Answer

1411

Try It 5.3.7

Simplify the complex rational expression by writing it as division: 341318+56

Answer

1023

We follow the same procedure when the complex rational expression contains variables.

Example 5.3.8

Simplify the complex rational expression by writing it as division: 1x+1yxyyx.

Solution
  1x+1yxyyx
Simplify the numerator. We will simplify the sum in the numerator and the difference in the denominator. 1x+1yxyyx
  Find the LCD in the numerator and denominator. =1yxy+1xyxxxyxyyxy
  Simplify. =yxy+xxyx2xyy2xy
 

Add the fractions in the numerator and subtract the fractions in the denominator.

We now have just one rational expression in the numerator and one in the denominator.

=y+xxyx2y2xy

Rewrite the complex rational expression as a division problem.

We write the numerator divided by the denominator.

=(y+xxy)÷(x2y2xy)
Divide the expressions. Multiply the first by the reciprocal of the second. =(y+xxy)(xyx2y2)
  Factor any expressions if possible. =xy(y+x)xy(xy)(x+y)
  Remove common factors. =xy(y+x)xy(xy)(x+y)
  Simplify. =1xy
Try It 5.3.9

Simplify the complex rational expression by writing it as division: 1x+1y1x1y.

Answer

y+xyx

Try It 5.3.10

Simplify the complex rational expression by writing it as division: 1a+1b1a21b2.

Answer

abba

We summarize the steps here.

How to Simplify a Complex Rational Expression by Writing It as Division
  1. Rewrite the complex rational expression as a division problem.
  2. Divide the expressions.
Example 5.3.11

Simplify the complex rational expression by writing it as division: n4nn+51n+5+1n5.

Solution
 

n4nn+51n+5+1n5

Simplify the numerator and denominator. Find common denominators for the numerator and denominator. =n(n+5)1(n+5)4nn+51(n5)(n+5)(n5)+1(n+5)(n5)(n+5)
Simplify the numerators. =n2+5nn+54nn+5n5(n+5)(n5)+n+5(n5)(n+5)
Subtract the rational expressions in the numerator and add in the denominator. =n2+5n4nn+5n5+n+5(n+5)(n5)
Simplify. (We now have one rational expression over one rational expression.) =n2+nn+52n(n+5)(n5)
Rewrite as fraction division. =n2+nn+5÷2n(n+5)(n5)
Multiply the first times the reciprocal of the second. =n2+nn+5(n+5)(n5)2n
Factor any expressions if possible.

=n(n+1)(n+5)(n5)(n+5)2n

Remove common factors. =n(n+1)(n+5)(n5)(n+5)2n
Simplify. =(n+1)(n5)2
Try It 5.3.12

Simplify the complex rational expression by writing it as division: b3bb+52b+5+1b5.

Answer

b(b+2)(b5)3b5

Try It 5.3.13

Simplify the complex rational expression by writing it as division: 13c+41c+4+c3.

Answer

3c+3

Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 7.4.2. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by LCDLCD we are multiplying by 1, so the value stays the same.

Example 5.3.14

Simplify the complex rational expression by using the LCD: 13+161213.

Solution
  13+161213

The LCD of all the fractions in the whole expression is 6.

Clear the fractions by multiplying the numerator and denominator by that LCD.

=6(13+16)6(1213)
Distribute. =613+616612613
Simplify.

=2+132

=31

=3

Try It 5.3.15

Simplify the complex rational expression by using the LCD: 12+15110+15.

Answer

73

Try It 5.3.16

Simplify the complex rational expression by using the LCD: 14+3812516.

Answer

103

We will use the same example as in Example 7.4.3. Decide which method works better for you.

Example 5.3.17

Simplify the complex rational expression by using the LCD: 1x+1yxyyx.

Solution
    1x+1yxyyx
Find the LCD of all fractions in the is complex rational expression. The LCD of all the fractions is xy. =1x+1yxyyx
Multiply the numerator and denominator by the LCD. Multiply both the numerator and denominator by xy. =xy(1x+1y)xy(xyyx)
Simplify the expression. Distribute.

=xy1x+xy1yxyxyxyyx

=y+xx2y2

  Simplify. =(y+x)(xy)(x+y)
  Remove common factors. =1xy
Try It 5.3.18

Simplify the complex rational expression by using the LCD: 1a+1bab+ba.

Answer

b+aa2+b2

Try It 5.3.19

Simplify the complex rational expression by using the LCD: 1x21y21x+1y.

Answer

yxxy

How to Simplify a Complex Rational Expression by Using the LCD
  1. Multiply the numerator and denominator by the LCD of all rational expressions.
  2. Simplify the expression.

Be sure to start by factoring all the denominators so you can find the LCD.

Example 5.3.20

Simplify the complex rational expression by using the LCD: 2x+64x64x236.

Solution
  2x+64x64x236
Find the LCD of all fractions in the complex rational expression.

The LCD is:

x236=(x+6)(x6).

Multiply the numerator and denominator by the LCD.

=(x+6)(x6)2x+6(x+6)(x6)(4x64(x+6)(x6))
Distribute in the denominator. =(x+6)(x6)2x+6(x+6)(x6)(4x6)(x+6)(x6)(4(x+6)(x6))
Simplify. =(x+6)(x6)2x+6(x+6)(x6)(4x6)(x+6)(x6)(4(x+6)(x6))
Simplify. =2(x6)4(x+6)4
To simplify the denominator, distribute and combine like terms. =2(x6)4x+20
Factor the denominator. =2(x6)4(x+5)
Remove common factors. =2(x6)22(x+5)
Simplify.

=x62(x+5)

Notice that there are no more factors common to the numerator and denominator.

Try It 5.3.21

Simplify the complex rational expression by using the LCD: 3x+25x23x24.

Answer

3(x2)5x+7

Try It 5.3.22

Simplify the complex rational expression by using the LCD: 2x71x+76x+71x249.

Answer

x+216x43

Be sure to factor the denominators first. Proceed carefully as the math can get messy!

Example 5.3.23

Simplify the complex rational expression by using the LCD: 4m27m+123m32m4.

Solution
  4m27m+123m32m4
Find the LCD of all fractions in the complex rational expression. The LCD is (m3)(m4).
Multiply the numerator and denominator by the LCD. =(m3)(m4)4(m3)(m4)(m3)(m4)(3m32m4)
Simplify. =(m3)(m4)4(m3)(m4)(m3)(m4)(3m3)(m3)(m4)(2m4)
Simplify. =43(m4)2(m3)
Distribute. =43m122m+6
Combine like terms. =4m6
Try It 5.3.24

Simplify the complex rational expression by using the LCD: 3x2+7x+104x+2+1x+5.

Answer

35x+22

Try It 5.3.25

Simplify the complex rational expression by using the LCD: 4yy+5+2y+63yy2+11y+30.

Answer

2(2y2+13y+5)3y

Example 5.3.26

Simplify the complex rational expression by using the LCD: yy+11+1y1.

Solution
  yy+11+1y1
Find the LCD of all fractions in the complex rational expression. The LCD is (y+1)(y1).
Multiply the numerator and denominator by the LCD. =(y+1)(y1)yy+1(y+1)(y1)(1+1y1)
Distribute in the denominator and simplify. =(y+1)(y1)yy+1(y+1)(y1)(1)+(y+1)(y1)(1y1)
Simplify. =(y1)y(y+1)(y1)+(y+1)

Simplify the denominator and leave the numerator factored.

=y(y1)y21+y+1

=y(y1)y2+y

Factor the denominator and remove factors common with the numerator. =y(y1)y(y+1)
Simplify. =y1y+1
Try It 5.3.27

Simplify the complex rational expression by using the LCD: xx+31+1x+3.

Answer

xx+4

Try It 5.3.28

Simplify the complex rational expression by using the LCD: 1+1x13x+1.

Answer

x(x+1)3(x1)

Key Concepts

  • How to simplify a complex rational expression by writing it as division.
    1. Simplify the numerator and denominator.
    2. Rewrite the complex rational expression as a division problem.
    3. Divide the expressions.
  • How to simplify a complex rational expression by using the LCD.
    1. Find the LCD of all fractions in the complex rational expression.
    2. Multiply the numerator and denominator by the LCD.
    3. Simplify the expression.

Glossary

complex rational expression
A complex rational expression is a rational expression in which the numerator and/or denominator contains a rational expression.

Practice Makes Perfect

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

1. 2aa+44a2a216

Answer

a42a

2. 3bb5b2b225

3. 5c2+5c1410c+7

Answer

12(c2)

4. 8d2+9d+1812d+6

5. 12+5623+79

Answer

1213

6. 12+3435+710

7. 231934+56

Answer

2057

8. 121623+34

9. nm+1n1nnm

Answer

n2+mmn2

10. 1p+pqqp1q

11. 1r+1t1r21t2

Answer

rttr

12. 2v+2w1v21w2

13. x2xx+31x+3+1x3

Answer

(x+1)(x3)2

14. y2yy42y4+2y+4

15. 22a+31a+3+a2

Answer

4a+1

16. 4+4b51b5+b4

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

17. 13+1814+112

Answer

118

18. 14+1916+112

19. 56+2971813

Answer

19

20. 16+4153512

21. cd+1d1ddc

Answer

c2+ccd2

22. 1m+mnnm1n

23. 1p+1q1p21q2

Answer

pqqp

24. 2r+2t1r21t2

25. 2x+53x5+1x225

Answer

2x103x+16

26. 5y43y+4+2y216

27. 5z264+3z+81z+8+2z8

Answer

3z193z+8

28. 3s+6+5s61s236+4s+6

29. 4a22a151a5+2a+3

Answer

43a7

30. 5b26b273b9+1b+3

31. 5c+23c+75cc2+9c+14

Answer

2c+295c

32. 6d42d+72dd2+3d28

33. 2+1p35p3

Answer

2p55

34. nn23+5n2

35. mm+54+1m5

Answer

m(m5)(4m19)(m+5)

36. 7+2q21q+2

In the following exercises, simplify each complex rational expression using either method.

37. 342712+514

Answer

1324

38. vw+1v1vvw

39. 2a+41a216

Answer

2(a4)

40. 3b23b405b+52b8

41. 3m+3n1m21n2

Answer

3mnnm

42. 2r91r+9+3r281

43. x3xx+23x+2+3x2

Answer

(x1)(x2)6

44. yy+32+1y3

Writing Exercises

45. In this section, you learned to simplify the complex fraction 3x+2xx24 two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?

Answer

Answers will vary.

46. Efraim wants to start simplifying the complex fraction 1a+1b1a1b by cancelling the variables from the numerator and denominator, 1a+1b1a1b. Explain what is wrong with Efraim’s plan.


This page titled 5.3: Simplify Complex Rational Expressions is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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