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Mathematics LibreTexts

7.1E: Exercises

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Practice Makes Perfect

In the following exercises, determine the values for which the rational expression is undefined.

Example 7.1E.49
  1. 2xz
  2. 4p16p5
  3. n3n2+2n8
Answer
  1. z=0
  2. p=56
  3. n=−4, n=2
Example 7.1E.50
  1. 10m11n
  2. 6y+134y9
  3. b8b236
Example 7.1E.51
  1. 4x2y3y
  2. 3x22x+1
  3. u1u23u28
Answer
  1. y=0
  2. x=12
  3. u=−4, u=7
Example 7.1E.52
  1. 5pq29q
  2. 7a43a+5
  3. 1x24

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

Example 7.1E.53

2xx1

  1. x=0
  2. x=2
  3. x=−1
Answer
  1. 0
  2. 4
  3. 1
Example 7.1E.54

4y15y3

  1. y=0
  2. y=2
  3. y=−1
Example 7.1E.55

2p+3p2+1

  1. p=0
  2. p=1
  3. p=−2
Answer
  1. 3
  2. 52
  3. 15
Example 7.1E.56

x+323x

  1. x=0
  2. x=1
  3. x=−2
Example 7.1E.57

y2+5y+6y21

  1. y=0
  2. y=2
  3. y=−2
Answer
  1. −6
  2. 203
  3. 0
Example 7.1E.58

z2+3z10z21

  1. z=0
  2. z=2
  3. z=−2
Example 7.1E.59

a24a2+5a+4

  1. a=0
  2. a=1
  3. a=−2
Answer
  1. −1
  2. 310
  3. 0
Example 7.1E.60

b2+2b23b4

  1. b=0
  2. b=2
  3. b=−2
Example 7.1E.61

x2+3xy+2y22x3y

  1. x=1, y=−1
  2. x=2, y=1
  3. x=−1, y=−2
Answer
  1. 0
  2. 34
  3. 154
Example 7.1E.62

c2+cd2d2cd3

  1. c=2, d=−1
  2. c=1, d=−1
  3. c=−1, d=2
Example 7.1E.63

m24n25mn3

  1. m=2, n=1
  2. m=−1, n=−1
  3. m=3, n=2
Answer
  1. 0
  2. 35
  3. 720
Example 7.1E.64

2s2ts29t2

  1. s=4, t=1
  2. s=−1, t=−1
  3. s=0, t=2

Simplify Rational Expressions

In the following exercises, simplify.

Example 7.1E.65

452

Answer

113

Example 7.1E.66

4455

Example 7.1E.67

5663

Answer

89

Example 7.1E.68

65104

Example 7.1E.69

6ab212a2b

Answer

b2a

Example 7.1E.70

15xy3x3y3

Example 7.1E.71

8m3n12mn2

Answer

2m23n

Example 7.1E.72

36v3w227vw3

Example 7.1E.73

3a+64a+8

Answer

34

Example 7.1E.74

5b+56b+6

Example 7.1E.75

3c95c15

Answer

35

Example 7.1E.76

4d+89d+18

Example 7.1E.77

7m+635m+45

Answer

75

Example 7.1E.78

8n963n36

Exercise 7.1E.79

12p2405p100

Answer

125

Example 7.1E.80

6q+2105q+175

Example 7.1E.81

a2a12a28a+16

Answer

a+3a4

Example 7.1E.82

x2+4x5x22x+1

Example 7.1E.83

y2+3y4y26y+5

Answer

y+4y5

Example 7.1E.84

v2+8v+15v2v12

Example 7.1E.85

x225x2+2x15

Answer

x5x3

Example 7.1E.86

a24a2+6a16

Example 7.1E.87

y22y3y29

Answer

y+1y+3

Example 7.1E.88

b2+9b+18b236

Example 7.1E.89

y3+y2+y+1y2+2y+1

Answer

y2+1y+1

Example 7.1E.90

p3+3p2+4p+12p2+p6

Example 7.1E.91

x32x225x+50x225

Answer

x−2

Example 7.1E.92

q3+3q24q12q24

Example 7.1E.93

3a2+15a6a2+6a36

Answer

a(a+5)2(a+3)(a2)

Example 7.1E.94

8b232b2b26b80

Example 7.1E.95

5c210c10c2+30c+100

Answer

c2(c5)

Example 7.1E.96

4d224d2d24d48

Example 7.1E.97

3m2+30m+754m2100

Answer

3(m+5)4(m5)

Example 7.1E.98

5n2+30n+452n218

Example 7.1E.99

5r2+30r35r249

Answer

5(r1)r+7

Example 7.1E.100

3s2+30s+723s248

Example 7.1E.101

t327t29

Answer

t2+3t+9t+3

Example 7.1E.102

v31v21

Example 7.1E.103

w3+216w236

Answer

w26w+36w6

Example 7.1E.104

v3+125v225

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

Example 7.1E.105

a55a

Answer

−1

Example 7.1E.106

b1212b

Example 7.1E.107

11cc11

Answer

−1

Example 7.1E.108

5dd5

Example 7.1E.109

122xx236

Answer

2x+6

Example 7.1E.110

205yy216

Example 7.1E.111

4v3264v2

Answer

48+v

Example 7.1E.112

7w219w2

Example 7.1E.113

y211y+249y2

Answer

y83+y

Example 7.1E.114

z29z+2016z2

Example 7.1E.115

a25a3681a2

Answer

a+49+a

Example 7.1E.116

b2+b4236b2​​​​​​​

Everyday Math

Example 7.1E.117

Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450 and $90,750, can be found by evaluating the formula 0.25x−4206.25, where x is income. The average tax rate for this income can be found by evaluating the formula 0.25x4206.25x. What would be the average tax rate for a single person earning $50,000?

Answer

16.5%

Example 7.1E.118

Work The length of time it takes for two people for perform the same task if they work together can be found by evaluating the formula xyx+y. If Tom can paint the den in x=45 minutes and his brother Bobby can paint it in y=60 minutes, how many minutes will it take them if they work together?

Writing Exercises

Example 7.1E.119

Explain how you find the values of x for which the rational expression x2x20x24 is undefined.​​​​​​​

Answer

Answers will vary, but all should reference setting the denominator function to zero.

Example 7.1E.120

Explain all the steps you take to simplify the rational expression p2+4p219p2.​​​​​​​

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This figure shows a table with four columns and five rows. The first row is a header row and each column is labeled. The first column header is labeled “I can…”, the second is labeled “Confidently”, the third is labeled “With some help”, and the fourth is labeled “No—I don’t get it!” In the first column under “I can”, the cells read “determine the values for which a rational expression is undefined,” “evaluate rational expressions,” “simplify rational expressions,” and “simplify rational expressions with opposite factors.” The rest of the cells are blank.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.


This page titled 7.1E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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