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Rational Expressions: Exercise Supplement

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.

The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.

This module contains the exercise supplement for the chapter "Rational Expressions".

Exercise Supplement

Rational Expressions ((Reference))

For the following problems, find the domain of each rational expression.

Exercise 1

9x+4

[ Show Solution ]

Exercise 2

10xx+6

Exercise 3

x+12x−5

[ Show Solution ]

Exercise 4

2a+37a+5

Exercise 5

3m2m(m−1)

[ Show Solution ]

Exercise 6

5r+69r(2r+1)

Exercise 7

ss(s+8)(4s+7)

[ Show Solution ]

Exercise 8

−11xx2−9x+18

Exercise 9

−y+512y2+28y−5

[ Show Solution ]

Exercise 10

1612a3+21a2−6a

For the following problems, show that the fractions are equivalent.

Exercise 11

−45,−45

[ Show Solution ]

Exercise 12

−38,−38

Exercise 13

−710,−710

[ Show Solution ]

For the following problems, fill in the missing term.

Exercise 14

−3y−5=y−5

Exercise 15

−6a2a+1=2a+1

[ Show Solution ]

Exercise 16

−x+1x−3=x−3

Exercise 17

−9−a+4=a−4

[ Show Solution ]

Exercise 18

y+3−y−5=y+5

Exercise 19

−6m−7−5m−1=6m+7

[ Show Solution ]

Exercise 20

−2r−57r+1=2r−5

Reducing Rational Expressions ((Reference))

For the following problems, reduce the rational expressions to lowest terms.

Exercise 21

126x+24

[ Show Solution ]

Exercise 22

164y−16

Exercise 23

5m+2510m2+15m

[ Show Solution ]

Exercise 24

7+21r7r2+28r

Exercise 25

3a2+4a5a3+6a2

[ Show Solution ]

Exercise 26

4x−4x2+2x−3

Exercise 27

5y+20y2−16

[ Show Solution ]

Exercise 28

4y3−12y4−2y2−3

Exercise 29

6a9−12a72a7−14a5

[ Show Solution ]

Exercise 30

8x4y8+24x3y94x2y5−12x3y6

Exercise 31

21y8z10w2−7y7w2

[ Show Solution ]

Exercise 32

−35a5b2c4d8−5abc3d6

Exercise 33

x2+9x+18x3+3x2

[ Show Solution ]

Exercise 34

a2−12a+352a4−14a3

Exercise 35

y2−7y+12y2−4y+3

[ Show Solution ]

Exercise 36

m2−6m−16m2−9m−22

Exercise 37

12r2−7r−104r2−13r+10

[ Show Solution ]

Exercise 38

14a2−5a−16a2+9a−6

Exercise 39

4a4−8a34a2

[ Show Solution ]

Exercise 40

5m210m3+5m2

Exercise 41

−6a−1−5a−2

[ Show Solution ]

Exercise 42

−r−5r−1

Multiplying and Dividing Rational Expressions ((Reference)) - Adding and Subtracting Rational Expressions ((Reference))

For the following problems, perform the indicated operations.

Exercise 43

x218 ⋅ 3x3

[ Show Solution ]

Exercise 44

4a2b315x4y5 ⋅ 10x6y3ab2

Exercise 45

x+6x−1 ⋅ x+7x+6

[ Show Solution ]

Exercise 46

8a−123a+3÷(a+1)24a−6

Exercise 47

10m4−5m24r7+20r3÷m16r8+80r4

[ Show Solution ]

Exercise 48

5r+7−3r+7

Exercise 49

2a3a−1−9a3a−1

[ Show Solution ]

Exercise 50

9x+74x−6+3x+24x−6

Exercise 51

15y−48y+1−2y+18y+1

[ Show Solution ]

Exercise 52

4a+3+6a−5

Exercise 53

7aa+6+5aa−8

[ Show Solution ]

Exercise 54

x+4x−2+x+7x−1

Exercise 55

2y+1y+4−y+6y+1

[ Show Solution ]

Exercise 56

x−3(x+2)(x+4)+2x−1x+4

Exercise 57

6a+5(2a+1)(4a−3)+4a+12a+1

[ Show Solution ]

Exercise 58

4x2+3x+2+9x2+6x+8

Exercise 59

6rr2+7r−18−−3rr2−3r+2

[ Show Solution ]

Exercise 60

y+3y2−11y+10−y+1y2+3y−4

Exercise 61

2a+516a2−1−6a+716a2−12a+2

[ Show Solution ]

Exercise 62

7y+46y2−32y+32+6y−102y2−18y+40

Exercise 63

x2−x−12x2−3x+2 ⋅ x2+3x−4x2−3x−18

[ Show Solution ]

Exercise 64

y2−1y2+9y+20÷y2+5y−6y2−16

Exercise 65

(r+3)4 ⋅ r+4(r+3)3

[ Show Solution ]

Exercise 66

(b+5)3 ⋅ (b+1)2(b+5)2

Exercise 67

(x−7)4÷(x−7)3x+1

[ Show Solution ]

Exercise 68

(4x+9)6÷(4x+9)2(3x+1)4

Exercise 69

5x+2x2+1x−4

[ Show Solution ]

Exercise 70

2y+4y2+5y−1

Exercise 71

y2+4y+4y2+10y+21÷(y+2)

[ Show Solution ]

Exercise 72

2x−3+4x2+x−1x−1

Exercise 73

3x+1x2+3x+2+5x+6x2+6x+5−3x−7x2−2x−35

[ Show Solution ]

Exercise 74

5a+3b8a2+2ab−b2−3a−b4a2−9ab+2b2−a+5b4a2+3ab−b2

Exercise 75

3x2+6x+1010x2+11x−6+2x2−4x+152x2−11x−21

[ Show Solution ]

Rational Equations ((Reference))

For the following problems, solve the rational equations.

Exercise 76

4x5+3x−115=2925

Exercise 77

6a7+2a−321=7721

[ Show Solution ]

Exercise 78

5x−16+3x+49=−89

Exercise 79

4y−54+8y+16=−6912

[ Show Solution ]

Exercise 80

4x−1+7x+2=43x2+x−2

Exercise 81

5a+3+6a−4=9a2−a−12

[ Show Solution ]

Exercise 82

−5y−3+2y−3=3y−3

Exercise 83

2m+5m−8+9m−8=30m−8

[ Show Solution ]

Exercise 84

r+6r−1−3r+2r−1=−6r−1

Exercise 85

8b+1b−7−b+5b−7=45b−7

[ Show Solution ]

Exercise 86

Solve z=x−x−x for s.

Exercise 87

Solve A=P(1+rt) for t.

[ Show Solution ]

Exercise 88

Solve 1R=1E+1F for E.

Exercise 89

Solve Q=2mns+t for t.

[ Show Solution ]

Exercise 90

Solve I=ER+r for r.

Applications ((Reference))

For the following problems, find the solution.

Exercise 91

When the same number is subtracted from both terms of the fraction 712, the result is 12. What is the number?

[ Show Solution ]

Exercise 92

When the same number is added to both terms of the fraction 1315, the result is 89. What is the number?

Exercise 93

When three fourths of a number is added to the reciprocal of the number, the result is 17316. What is the number?

[ Show Solution ]

Exercise 94

When one third of a number is added to the reciprocal of the number, the result is −12790. What is the number?

Exercise 95

Person A working alone can complete a job in 9 hours. Person B working alone can complete the same job in 7 hours. How long will it take both people to complete the job working together?

[ Show Solution ]

Exercise 96

Debbie can complete an algebra assignment in 34 of an hour. Sandi, who plays her radio while working, can complete the same assignment in 114 hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?

Exercise 97

An inlet pipe can fill a tank in 6 hours and an outlet pipe can drain the tank in 8 hours. If both pipes are open, how long will it take to fill the tank?

[ Show Solution ]

Exercise 98

Two pipes can fill a tank in 4 and 5 hours, respectively. How long will it take both pipes to fill the tank?

Exercise 99

The pressure due to surface tension in a spherical bubble is given by P=4Tr, where T is the surface tension of the liquid, and r is the radius of the bubble.
 (a) Determine the pressure due to surface tension within a soap bubble of radius 12 inch and surface tension 22.
 (b) Determine the radius of a bubble if the pressure due to surface tension is 57.6 and the surface tension is 18.

[ Show Solution ]

Exercise 100

The equation 1p+1q=1f relates an objects distance p from a lens and the image distance q from the lens to the focal length f of the lens.
 (a) Determine the focal length of a lens in which an object 8 feet away produces an image 6 feet away.
 (b) Determine how far an object is from a lens if the focal length of the lens is 10 inches and the image distance is 10 inches.
(c) Determine how far an object will be from a lens that has a focal length of 178 cm and the object distance is 3 cm away from the lens.

Dividing Polynomials ((Reference))

For the following problems, divide the polynomials.

Exercise 101

a2+9a+18 by a+3

[ Show Solution ]

Exercise 102

c2+3c−88 by c−8

Exercise 103

x3+9x2+18x+28 by x+7

[ Show Solution ]

Exercise 104

y3−2y2−49y−6 by y+6

Exercise 105

m4+2m3−8m2−m+2 by m−2

[ Show Solution ]

Exercise 106

3r2−17r−27 by r−7

Exercise 107

a3−3a2−56a+10 by a−9

[ Show Solution ]

Exercise 108

x3−x+1 by x+3

Exercise 109

y3+y2−y by y+4

[ Show Solution ]

Exercise 110

5x6+5x5−2x4+5x3−7x2−8x+6 by x2+x−1

Exercise 111

y10−y7+3y4−3y by y4−y

[ Show Solution ]

Exercise 112

−4b7−3b6−22b5−19b4+12b3−6b2+b+4 by b2+6

Exercise 113

x3+1 by x+1

[ Show Solution ]

Exercise 114

a4+6a3+4a2+12a+8 by a2+3a+2

Exercise 115

y10+6y5+9 by y5+3

[ Show Solution ]

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