1

MATH 521B WORKBOOK

RADICALS

AND

RATIONAL EXPRESSIONS

2

Unit 1 - Radicals

Estimation of Radicals

Estimate, then find an approximate value, to the nearest hundredth.1. 2. 3. 4. 87 6680 82 55. 0 22.5. 6. 0 0038. 2 006.

Estimate, then find an approximate value, to the nearest tenth.7. 8. 90 56− 9 189

Give a decimal approximation to the nearest hundredth fo each radical.9. 10. 11. 12. 13. 14. 39 870 4503 73003 1444 9004

Use decimal approximations to arrange the following in order from least to greatest.15. , , 17 3 2 2 516. 8, , , 2 15 3 7 6217. , , , 5 3 74 4 5 6 2

Simplifying Radicals

Simplify.1. 2. 3. 4. 5. 6. 12 45 24 200 44 187. 8. 128 1259. a) b) c) d) e) f) 163 323 543 813 324 644

Simplify.10. 11. 2503 1353

12. a) b) c) d) 32 323 324 325

Simplify. Assume that each radical represents a real number.

13. 14. 15. 16. 18 2x 12 5x 375 53 a 16 43 c

3

17. Express the exact area of the triangle in simplest radical form.

18. A square has an area of 675 cm2. Express the side length in simplest radical form.

Write as an entire radical.19. 20. 21. 22. 23. 24. 2 3 4 2 3 10 3 5 2 7 6 8

Without using a calculator, arrange the following in order from least to greatest.25. , , , 3 5 2 11 4 3 5 226. , , , 6 2 3 7 8 2 1527. , , , 5 5 4 7 3 14 2 30

Operations with Radicalsa) add or subtract

Simplify.1. 2. 3. 12 27+ 20 45+ 18 8−4. 5. 6. 50 98 2+ − 75 48 27+ + 54 24 72 32+ + −7. 8. 9. 28 27 63 300− + + 8 7 2 28+ 3 50 2 32−10. 11. 12. 5 27 4 48+ 3 8 18 3 2+ + 5 2 45 3 20+ −13. 14. 4 3 3 20 2 12 45+ − + 3 48 4 8 4 27 2 72− + −15. Express the volume of the rectangular prism in simplest radical form.

15 2−

5 2 2 3− 5 2 2 3+

4

Simplify. If no simplification is possible, say so.16. 17. 18. 19. 50 18+ 45 20− 3 12 48− 27 2 75+20. 21. 22. 5 2 2 5− 7 3 3 7− 6 36 216+ +23. 24. 25. 5 25 125+ + 50 63 32+ − 18 24 54+ −26. 27. 54 40 163 3 3+ + 24 56 813 3 3− +

Operations with Radicalsb) Multiply and divide

Simplify.1. 2. 3. 2 10× 3 6× 15 5×4. 5. 6. 7 11× 4 3 7× 3 6 3 6×7. 8. 9. 2 2 3 6× 2 5 3 10× 3 3 4 15×10. 11. 12. 4 7 2 14× 6 3 2× × 2 7 3 1 7× ×

Expand and simplify.

13. 14. 15. 2 10 4( )+ 3 6 1( )− 6 2 6( )+

16. 17. 18. 2 2 3 6 3( )− 2 3 4( )+ 3 2 2 6 10( )+

19. 20. 21. ( )( )5 6 5 3 6+ + ( )( )2 3 1 3 3 2− + ( )( )4 7 3 2 2 7 5 2− +

22. 23. 24. ( )3 3 1 2+ ( )2 2 5 2− ( )( )2 3 2 3+ −

25. 26. ( )( )6 2 6 2− + ( )( )2 7 3 5 2 7 3 5+ −27. Write and simplify an expression for the area of the square below.

8 5−

5

45° - 45° - 90° triangle28. Find the length of the hypotenuse in each isosceles right triangle. Write your answer as a mixedradical in simplest form. a) b) c)

2 2 h 3 h 5

h 3 5

29. What is the relationship between the length of the hypotenuse and the length of each leg in a rightisosceles triangle?

30. Find the lengths of the indicated sides in each isosceles right triangle.a) b) c) d)

c d 7 k h 10 a 6 2

8 h k a

31. If the length of each leg of a isosceles right triangle is 1, what is the length of the hypotenuse?

32. If the length of each leg of a isosceles right triangle is s, what is the length of the hypotenuse in termsof s ?

30° - 60° - 90° triangle33. Find the lengths of the indicated sides in each triangle. Write radical answers as mixed radicals insimplest form.a) b) c) e)

d

4 b 8 k 2 x 10 e

a

g y

34. What is the relationship between the length of the hypotenuse and the length of the side opposite the30° angle?

35. What is the relationship between the length of the side opposite the 60° angle and the length of the

6

side opposite the 30° angle?

36. Find the lengths of the indicated sides in each 30° - 60° - 90° triangles.a) b) c)

k 8 h3 3

a h h

a 6

37. If the length of the hypotenuse is 1, what are the lengths of the other two sides?

38. If the length of the hypotenuse is s, what are the lengths of the other two sides in terms of s?

The Equilateral triangle39. Given equilateral triangle DEF, finda) the altitude, ab) the area, A

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40. Given equilateral triangle XYZ, finda) the altitude, ab) the area, A

41. Given equilateral triangle PQR, express a) the altitude, a, in terms of sb) the area, A, in terms of s

42. Use your formula from question 3b) to find the area of each ofthe following triangles.a) b) c)

7 10

6 6 10

7 7 6 10

Problem Solving43. An equilateral triangle has an area of 36 cm2 . Calculate the side length and the height of the triangle,to the nearest tenth of a centimetre.

44. A regular hexagon has an area of 600 cm2. Calculate the length of each side, to the nearest tenth of acentimetre.

45. A 30° - 60° - 90° triangle has an area of 20 cm2. Calculate the length of each side, to the nearest tenthof a centimetre.

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46. The equilateral triangle is inscribed in a circle. The circle has an area of 64B. State the exact area ofthe triangle as a mixed radical in simplest form.

47. a) Find the area of triangle ABC, in square centimetres.

b) Triangle DEF is similar to triangle ABC. Find the area of triangle DEF, in square centimetres.

Simplify.

48. 49. 50. 2706

963

30 42⋅

51. 52. 53. 35 21⋅ 6 23

⋅ 15 35

⋅

54. 55. 56. 57. ( )2 7 2 ( )3 6 2 45 123 3⋅ 20 143 3⋅

Simplify.

9

58. 59. 60. 2 8 10( )+ 3 12 24( )− 15 3 2 5( )+

61. 62. 63. 7 3 14 21( )− 2 3 48 5 12( )− 3 5 5 2 75( )+

64. 65. 5 200 163 3 3( )− 40 25 2 53 3 3( )+

Simplify.

66. 67. 68. ( )( )3 7 3 7+ − ( )( )5 2 5 2+ − ( )7 1 2+

69. 70. 71. ( )5 2 2+ ( )( )1 2 3 2+ + ( )( )6 3 4 3− +

72. 73. 74. ( )7 2 2− ( )3 11 10 2− ( )( )3 4 3 2 3+ −

75. 76. 77. ( )( )5 2 3 2 2− − ( )( )11 7 11 7− + ( )( )13 3 13 3− +

78. 79. 80. ( )( )5 3 8 2 3+ − ( )( )3 2 6 4 5 6+ − ( )2 5 7 2+

81. 82. 83. ( )3 2 6 2+ ( )( )2 3 5 2 3 5+ − ( )( )3 7 2 5 3 7 2 5− +

84. 85. 86. ( )6 15 2− ( )2 5 10 2−2 7 1

32 7 1

3+

⋅−

87. 88. ( )( )5 6 3 2 6 4 3+ − ( )( )3 5 2 15 4 3 3 15+ −

Rationalizing denominators

Simplify.

1. 2. 3. 4. 5. 13

37

56

5 52 3

2 218

6. 7. 8. 9. 23

4 28

4 72 14

3 64 10

Simplify.

10. 11. 12. 13. 1

2 2+3

5 1−2

6 3−2

6 3+

14. 15. 16. 17. 3

5 2−

33 2+

2 62 6 1+

2 12 1−+

18. 19. 2 56 10+−

2 7 53 7 2 5

+−

10

Simplify.

20. 21. 22. 23. 2

3 2+5

10 5−3 3

2 6 3 2−

4 107 2 10−−

24. 25. 6 2 36 2 3+−

3 7 2 23 2 2 7

−−

26. If a rectangle has an area of 4 square units and a width of units, what is its length in7 5−simplest radical form?

27. Express the ratio of the area of the larger circle to the area of the smaller circle in simplest radicalform.

2 3+2 3−

11

Simplify.

29. 30. 31. 32. 33. 34. 43

95

42

63

29

36036

3

3

35. 17550

3

3

36. a) b) c) d) 38

38

338

438

5

Simplify. If no simplification is possible, say so.

37. 38. 39. 275

35

−752

32

−23

32

+

40. 41. 42. 52

25

+ 4 12

3 3+ 16 14

3 3−

12

Rational Exponents

Write in radical form.

1. 2. 3. 4. 5. 6. 213 37

32 x

12 a

15 6

43 6

34

7. 8. 9. 10. 11. 12. 712

−9

15

−x−

37 b

−65 ( )3

12x 3

12x

Write using exponents.

13. 14. 15. 16. 17. 18. 7 34 − 113 a 25 643 ( )b3 4

19. 20. 21. 22. 23. 24. 1x

13 a

15 4( )x

2 33 b 3 5x 5 34 t

Evaluate.

25. 26. 27. 28. 29. 30. 412 125

13 16

14

−( )−32

15 250 5. ( )−

−27

13

31. 32. 33. 34. 35. 36. ( )6416

−0 04

12. 810 25. 0 001

13. 4

9

12⎛

⎝⎜⎞⎠⎟

−−

⎛⎝⎜

⎞⎠⎟

278

13

Evaluate.

37. 38. 39. 40. 41. 42. 823 4

32 92 5. 81

34 16

34

−( )−32

25

43. 44. 45. 46. 47. 48. ( )−−

853 ( )−

−27

23 1

53 ( )−

−1

85 100

9

32⎛

⎝⎜⎞⎠⎟

278

23⎛

⎝⎜⎞⎠⎟−

Simplify.

49. a) b) 912 9

12

−

50. a) b) 813 8

13

−

51. a) b) 823 8

23

−

52. a) b) 1614 16

14

−

13

53. 54. 55. 56. 2713

−27

23 4

32 25

12

−

Simplify.

57. 58. 59. 60. 61. 8112 49

12

−4

32 16

34 ( )−

−125

13

62. 63. 64. 65. 66. 4 0 5− . − 823 ( )5

13 3− ( )16 5

120− ( )9 16

12

12 2+

Express in simplest radical form.

67. 68. 69. 70. 4 43 3⋅42

3

632 410 8÷ 27 94 8⋅

Radical Equations

Solve and check.

1. 2. 3. 4. x = 5 x − =2 0 x + =3 0 y + =1 2

5. 6. 7. 8. 3 4− =m 1 2 0− + =x 2 1 3x + = 4 3 2 0− − =x

9. 10. 11. 12. x − + =3 6 2 z3

1 2 4+ + = 4 3 6( )x + = 05 3 2 2 1. ( )x − + =

13. 14. 2 1 2 8x − − = − + + =3 2 4 1x

Solve. If an equation has no real solution, say so.15. 16. 17. 18. 4 3 5x − = 3 1 7n + = 3 5 13t − = 7 4 3+ =a

19. 20. 21. 22. 2 7 52x − = 5 1 92y + = 3 1 43 m + = 2 5 33 w − =

23. 24. 2 5 33 d + = 7 9 43− =c

Equations with Rational Exponents

Give the power to which you would raise both sides of each equation in order to solve the equation.

1. 2. 3. 4. x12 9= x

23 4= x

−=

13 2 x

− −=34 18

Tell what steps you would use to solve each equation.

5. 6. 7. 8. 3 614x = 5 40

32x

−= ( )x − =−3 1

42 ( )5 3

12x

−=

14

Solve each equation.

9. a) b) a34 8= ( )3 1 8

34x + =

10. a) b) y−

=12 6 ( )3 6

12y

−=

11. a) b) 2 1012y

−= ( )2 10

12y

−=

12. a) b) ( )9 423t

−= 9 4

23t

−=

13. ( )8 413− =y

14. ( )3 1 14

23n − =

−

15. ( )x 2234 25+ =

16. ( )x 2129 5+ =

Exponential Equations

Solve.

1. 2. 3. 4. 3 19

x = 25 125x = 4 18

x = 36 6x =

Solve. If an equation has no solution, say so.

5. 6. 7. 8. 3 127

x = 5 125x = 8 22+ =x 4 81− =x

9. 10. 11. 12. 27 32 1x− = 49 7 72x− = 4 162 5 1x x+ += 3 95 4− + =( )x x

13. 14. 15. 25 52 6x x= + 6 361 1x x+ −= 10 1001 4x x− −=

15

Answers for Unit 1 - Radicals

Estimation of Radicals

1. 9; 9.332. 80; 81.733. 9; 9.094. 0.5; 0.475. 0.06; 0.066. 2; 2.07. 1; 1.4168. 126; 123.7

9. 6.2410. 29.5011. 7.6612. 19.4013. 3.4614. 5.48

31. , , 32. , , ,8 33. , , , 17 3 2 2 5 2 15 62 3 7 6 2 74 5 3 4 5

Simplifying Radicals

1. 2. 3. 4. 5. 6. 3 2 3 5 2 6 10 2 2 11 3 27. 8. 8 2 5 5

9. a) b) c) d) e) f) 2 23 2 43 3 23 3 33 2 24 2 44

10. 11. 5 23 3 53

12. a) b) c) d) 24 2 2 43 2 24

13. 14. 15. 16. 3 2x 2 32x x 5 3 23a a 2 23c c

17. 18. 152

15 3

16

19. 20. 21. 22. 23. 24. 12 32 90 45 28 288

25. , , , 2 11 3 5 4 3 5 226. , , 8, 2 15 3 7 6 227. , , , 4 7 2 30 5 5 3 14

Operations with Radicalsa) add or subtract

1. 2. 3. 5 3 5 5 24. 5. 6. 11 2 12 3 5 6 2 2+7. 8. 9. 5 7 7 3+ 12 7 7 210. 11. 12. 31 3 12 2 513. 14. 9 5 24 3 20 2−15. 38 15 38 2−

16. 17. 18. 19. 20. not possible8 2 5 2 3 13 321. not possible 22. 23. 24. 6 7 6+ 5 6 5+ 2 3 7+25. 26. 27. 3 2 6− 5 2 2 53 3+ 5 3 2 73 3−

17

Operations with Radicalsb) Multiply and divide

1. 2. 3. 4. 5. 2 5 3 2 5 3 77 4 216. 7. 8. 9. 10. 54 12 3 30 2 36 5 56 211. 12. 426

13. 14. 15. 16. 2 5 4 2+ 3 2 3− 2 3 6+ 12 3 2 6−17. 18. 19. 20. 6 4 2+ 12 3 6 5+ 23 4 30+ 16 3+21. 22. 23. 24. 26 14 14+ 28 6 3+ 13 4 10− 125. 26. 4 − 1727. 13 4 10−

28. a) b) c) 2 2 3 2 5 229. The length of the hypotenuse is times the length of a leg.2

30. a) b) k h= =7 7 2, k h= =10 10 2,c) d) a b= = 6 c d= = 4 2

31. 32. 2 s 2

33. a) b) c) d) a b= =2 2 3, x y= =5 3 5, d e= =4 3 4, g k= =1 3,

34. The length of the side opposite the 30° angle is half the length of the hypotenuse.

35. The length of the side opposite the 60° angle is times the length of the side opposite the 30°3angle.

36. a) b) c) h a= =12 6 3, h k= =16 8 3, a h= =3 6,

37. 12

32

,

38. s s2

32

,

39. a) b) 2 3 4 3

18

40. a) b) 3

23

4

41. a) b) s 3

2s2 3

4

42. a) b) c) 9 3 25 3 49 34

Problem Solving43. 9.1 cm, 7.9 cm44. 15.2 cm45. 4.8 cm, 8.3 cm, 9.6 cm46. 48 3

47. a) 16 cm2 b) 32 cm2

48. 49. 50. 51. 52. 53. 3 5 4 2 6 35 7 15 2 354. 28 55. 54 56. 57. 3 203 2 353

58. 59. 60. 61. 4 2 5+ 6 6 2− 3 5 10 3+ 21 2 7 3−62. 63. − 36 15 30 15+64. 65. 10 2 103− 10 4 253+

66. 2 67. 23 68. 69. 70. 71. 8 2 7+ 9 4 5+ 5 4 2+ 21 2 3+72. 73. 74. 75. 9 2 14− 109 6 110− − +6 5 3 19 13 2−76. 4 77. 10 78. 79. 34 2 3− − −48 7 680. 81. 82. 83. 27 4 35+ 24 12 3+ 7 4384. 85. 86. 87. 21 6 10− 30 20 2− 3 60 60 2 12 3 12 6− + −88. 12 15 45 3 24 5 90− + −

Rationalizing denominators

1. 2. 3. 4. 5. 3

3217

306

5 156

23

6. 7. 8. 9. 2 3

32 2 3 15

20

19

10. 11. 12. 13. 2 2

2− 3 5 3

4+ 2 3 3 2

3+−

2 6 2 33−

14. 15. 16. 17. 5 2+ 3 6−24 2 6

23− 3 2 2−

18. 19. 2 3 2 5 30 5 2

4+ + +

−52 7 35

43+

20. 21. 22. 23. 2 3 2 2− 2 1+ 6 2 3 62+ 8 10

9+

24. 25. − −3 2 2 30 5 1410

+−

26. 27. 28. a) b) 2 7 2 5+ 97 56 3+ 4 13 8+ 3 2 2 5+

29. 30. 31. 32. 33. 34. 2 3

33 5

52 2 2 3 6

3

3 453

3

35. 36. a) b) c) d) 5 28

10

3 64

32

3 62

4 122

5

37. 38. 39. 2 15

52 6 5 6

6

40. 41. 2 42. 7 10

103 2

2

3

Rational Exponents

1. 2. 3. 4. 5. 6. 23 ( )37 3 x a5 ( )63 4 ( )63 3

7. 8. 9. 10. 11. 12. 17

195

137 x

165 b

3x 3 x

13. 14. 15. 16. 17. 18. 712 34

12 ( )−11

13 a

25 6

43 b

43

19. 20. 21. 22. 23. 24. x−

12 a

−13 x

−45 2

13 b 3

12

52x 5

14

34t

25. 26. 27. 28. 29. 30. 2 5 12

− 2 5 −13

20

31. 32. 33. 34. 35. 36. 12

0 2. 3 01. 23

32

37. 38. 39. 40. 41. 42. 4 8 243 27 18

4

43. 44. 45. 46. 47. 48. −1

3219

1 1 100027

49

49. a) 3 b) 13

50. a) 2 b) 12

51. a) 4 b) 14

52. a) 2 b) 12

53. 54. 9 55. 8 56. 13

15

57. 9 58. 59. 8 60. 8 61. 17

−15

62. 63. !4 64. 65. 66. 4912

15

12

67. 68. 69. 70. 2 23 2 24 3

Radical equations

1. 25 2. 4 3. !3 4. 3 5. !13 6 . No solution

7. 4 8. 0 9. no solution 10. 9 11. 6 12. No solution

13. 26 14. !1

15. 7 16. 16 17. 36 18. no solution 19. !4, 4 20. !4, 4

21. 21 22. 16 23. !4 24. 3

21

Equations with Rational Exponents

1. 2 2. 3. 4. 32

− 3 −43

9. a) 16 b) 5

10. a) b) 136

1108

11. a) b) 125

1200

12. a) b) 1

72278

13. !56 14. 3 15. ±11 16. ±4

Exponential Equations

1. 1 2. 3. 4. 32

12

14

5. !3 6. 7. 8. 9. 10. 32

−53

−12

23

114

11. No solution 12. 13. 14. 3 15. 3−59

2

22

Unit 5 - Rational expressions

Rational Expressions

1. For which values of x are the following rational expressions not defined?

a) b) c) d) e) 2x yx y−−

43

xx y+

33x

xx

2

3 8−x x

x

2

4

3 111

+ −−

f) 3 5 2

4 9

2 2

2 2

x xy yx y+ +

−

Find (a) the domain of each function and (b) its zeros, if any.

2. 3. 4. f t tt t

( ) = −−

2

2

99

g x x xx

( ) = +−

3

2

24

F x x x( ) ( )( )= − − −4 3 116 1

5. 6. 7. h y y y( ) ( )( )= − + −3 38 2 g t t tt t

( ) = + −−

2 3 94

2

3 G s s ss

( )( )

=+ −−

4 15 42 1

2

2

8. 9. f x x x xx x

( ) = − + −+ −

3 2

4 2

2 22

h t t t tt t t

( ) = + − −− + −

3 2

3 2

4 41

Simplifying Rational Expressions

Reduce to lowest terms. State any restrictions on the variables.

1. 2. 3. 4. 4 45 5

x yx y++

6 366

tt−−

5 103 6xx−−

2 22 2

2

2

x xx x

−+

5. 5 102 42

xy xy y++

23

Express in simplest equivalent form.

6. 7. 8. y y

y

2 10 255

+ ++

rr

2 45 10

−+

xx y xy

2

2

92 6

−−

9. 10. 3 8 4

6 4

2

2

t tt t− +−

5 3 23 3

2 2

2

x xy yx xy+ −

+Simplify.

11. 12. 13. a a

a a

2

2

129 20− −− +

y yy

2

2

8 1525

− +−

n nn n

2

2

26

− −+ −

14. 15. 2 1

3 2

2

2

t tt t

− −− +

6 13 68 6 9

2

2

x xx x− +− −

16. The area of a Saskatchewan flag can be represented by the polynomial x2 +3x + 2 and its width by x + 1. a) Write a rational expression that represents the length.b) Write the expression in simplest form.c) If x represents 1 unit of length, what is the ratio length : width for a Saskatchewan flag?

17. The length of an edge of a cube is x + 1 . Write and simplify a rational expression that represents theratio of the volume to the surface area.

18. The diagrams show the numbers of asterisks in the first 4 diagrams of two patterns.

Pattern 1

Pattern 2

24

19. (cont’d.)a) For pattern 1, express the number of asterisks in the nth diagram in terms of n.b) For the pattern 2, the number of asterisks in the nth diagram is given by the binomial product (n + __)(n + __), where the blanks represent whole numbers. Replace the blanks in the binomial productwith their correct values.c) Divide your polynomial from part b) by your expression from part a).d) Use your results from part c) to calculate how many times more asterisks there are in the 10th diagramof pattern 2 than there are in the 10th diagram of pattern 1.e) If a diagram in pattern 1 has 20 asterisks, how many asterisks are in the corresponding diagram ofpattern 2?f) If a diagram in pattern 2 has 1295 asterisks, how many asterisks are in the corresponding diagram ofpattern 1?

Simplify.

20. 21. 22. 5 15

10

2

2

x xx− u u

u u

2

2

2− −+

( )( )p q q p− − −1

23. 24. 25. s tt s

2 2

2

−−( )

x xx x

2

2

5 67 12− +− +

6 5 11 6

2

2

y yy y− +

− −

26. 27. ( )( )r r r2 25 4 4− + − − x xx x

2 2 82 4

+ −− +( )( )

28. 29. x x xx x x

3 2

3 2

11

+ − −− − +

x x y xy yx y

3 2 2 3

4 4

− + −−

30. 31. s t

s s t t

4 4

4 2 2 42−

− +x x y xy y

x y

4 3 3 4

4 4

+ − −−

Multiplying or Dividing Rational Expressions

Simplify. State any restrictions on the variables.

1. 2. 3. 16

9384 2

5 4

2 2

abx y

x ya b

×5

61092

2

3 2

xyx y

xyx y

÷ 6 23

3 4x y xy÷−

4. 5. 6. 3

44

6xx

−×

− 5 21

110

( )yy

y−+

×+ 4

382 2a b

a bab

a b( )+÷−

+

7. 8. x xx x

x xx x

2

2

2

2

5 66 5

309 18

+ +− +

×+ −+ +

m mm m

m mm m

2

2

2

2

3 45

7 122 15

− −+

÷− ++ −

9. 10. x xy

x xy yx xy y

x y

2

2 2

2 2

2 2

342

10 219

+− −

×− +

−12 19 5

4 92 33 1

2

2

a aa

aa

− +−

×−−

25

11. 12. 12 5 28 2 21

12 68 2 15

2

2

2

2

w ww w

w ww w

− −+ −

÷+ −

− −3 14 5

4 7 36 2

8 2 3

3 2

2

2

2

x x xx x

x xx x

+ −+ +

÷−

+ −

13. 14.

3 62 2

212

nn

nn

+++−

6 37 69 18

12 16 35 12 9

2

2

2

2

x xx x

x xx x

− +− ++ −− −

26

Simplify. Write answers without negative or zero exponents.

20. 21. 22. 5

36

10

3

2

xx−

⋅− 8

329

2t t÷

x xy yx

2 1 2

4 62

⋅ ⎛⎝⎜

⎞⎠⎟

⋅−

23. 24. 25. 445

827

910

2rs sr

rs÷ ÷

x xx

xx

( )( )

( )−−

÷−−

12

122

2 4 14

22 1

2

2

uu

uu

−−

⋅−−

26. 27. x

x xx x

x

2

2

2

2

42 5 2

2 3 24 1

−− +

÷− −

−

p qp q

p q

4 4

2

2 2

−++

( )

28. u v

u vu v u uv v

uv u v

2 2 2

2 2

2+

÷ + ⋅+ +

−( )

29. 3 23 2

3 7 63 2

33 2

2 2

2 2

2 2

2 2

x xy yx xy y

x xy yx xy y

x yx y

+ −− −

÷+ −− −

÷++

27

Adding or Subtracting Rational Expressions

Simplify. In each of the following, state any restrictions on the variables.

1. 2. 3. 2 3

23 4

7m m+

++ y y−

−−5

62 3

44 1

63 2

22 1

3t t t−

++

−+

28

Simplify. In each of the following, state any restrictions on the variables.

7. 8. 9. 2

13

2x x++

+3 5

1x x+

−2

23

2x

xx

x−−

+

10. 11. 12. y

y y2 164

4−−

+a

a aa

a2 7 122

3− +−

−6

2 13

6 5 12n n n−−

− +

13. 14. 14 4

342 2x x x+ +

−−

49 18

211 302 2

mm m

mm m− +

+− +

15. 3

4 92

4 12 92 2

yy

yy y−

−− +

16. An RCMP patrol boat left Tofino on Vancouver Island and travelled for 45 km along the coast at aspeed of s kilometres per hour.a) write an expression that represents the time taken, in hours.b) The boat returned to Tofino at a speed of 2s kilometres per hour. Write an expression that representsthe time taken, in hours.c) Write and simplify an expression that represents the total time, in hours, the boat was at sea.d) If s represents 10 km/h, for how many hours was the boat at sea?

17. Simplify. In each of the following, state any restrictions on the variables.

a) b) c) 2 1

2 1

+

−

x

x

y

y

+

−

1412

1 1

2 22−

−

m

m

d) e) f)

t

tt

21

14

12

+

++

n n

n

2

2

92

3

9

++

−

x x

x x

2

2

5 32

8 33

−+

−+

( )

( )

18. Write two rational expressions with binomial denominators and with each of the following sums. Compare your answers with a classmates’s.

a) b) c) d) 5 81 2x

x x+

+ +( )( )5 5

6 13 62

xx x

−− +

xx x

2 31 3−

− −( )( )4

4 9

2

2

xx −

29

Simplify.

19. 20. 21. 5

161216

316

− +12

37

514

− +t t+

+−2

34

6

22. 23. x x+

−+3

52 1

1012 1

112 2s s s+ +

−−

24. 25. 35 6

242 2x x x− +

+−

14 4 1

14 12 2t t t− +

+−

26. 27. 1

2 31

42 2 2 2u uv v u v− ++

−3

4 12 91

2 32 2 2x xy y xy y− ++

−

Rational Equations

Solve each open sentence.

1. 2. 3. x9

16

23

+ =2 1

62

413

t t−=

++

z z2

3 61− =

4. 5. x x x( )+

−+

=1

51

613

y y y y( ) ( )2 14

310

25

−+ =

+

6. Pump A can unload a ship in 30 h and pump B can unload the ship in 24 h. Because of an approachingstorm, both pumps were used. How long did both pumps take to unload the ship?

7. An old conveyor belt takes 21 h to move one day’s coal output from a mine to a rail line. A new beltcan do the same job in 15 h. How long does it take when both belts are used at the same time?

8. A river boat paddled upstream at 12 km/h, stopped for 2 h of sightseeing, and paddled back at 18 km/h. How far upstream did the boat travel if the total time for the trip, including the stop, was 7 h?

Solve and check. If an equation has no solution, say so.

9. 10. 1

21

24

42y y y−+

+=

−3

11

21

22x x x x+−

−=

− −

11. 12. 6 2

11

t t−

−=

uu u−

++

=2

302

9

13. 14. 2

1 362 32x

xx x x−

−+

=+ −

56

2 322u u

uu+ −

= −−−

30

15. A town’s old street sweeping machine can clean the streets in 60 h. The old sweeper together with anew sweeper can clean the streets in 15 h. How long would it take the new sweeping machine to do thejob alone?

16. The intake pipe can fill a certain tank in 6 h when the outlet pipe is closed, but with the outlet pipeopen it takes 9 h to fill the tank. How long would it take the outlet pipe to empty a ful tank?

17. An excursion boat travels 35 km upstream and then back again in 4 h 48 min. If the speed of the boatin still water is 15 km/h, what is the speed of the current?

18. Members of a ski club contributed equally to obtain $1800 for a holiday trip. When 6 members foundthat they could not go, their contributions were refunded and each remaining member then had to pay $10more to raise the $1800. How many went on the trip?

31

Answers for Unit 5 - Rational expressions

1. a) x = y b) c) x = 0 d) x = 2 e) x = ±1 f) x y= −

3x y= ±

32

2. Reals except 0 and 9; ±3 3. Reals except ±2; 0 4. Reals except 1; ±2

5. Reals except !; 2 6. Reals except 0,±2; !3, 7. Reals except ; 32

12

14

4,−

8. Reals except ±1; 2 9. Reals except 1; ±, !4

Simplifying Rational Expressions

1. 2. 3. 4. 45

, x y≠ 6 6, t ≠ 53

2, x ≠ xx

x−+

≠ −11

0 1, ,

5. 6. 7. 8. 52

0 2xy

y, .≠ − y y+ ≠ −5 5, r r−≠ −

25

2, xxy

x y+≠ ≠

32

0 3 0, , ,

9. 10. 11. t

tt−≠

22

0 23

, , 5 23

0x yx

x y−≠ −, , a

aa+

−≠

35

4 5, ,

12. 13. 14. yy

y−+

≠ ±35

5, nn

n++

≠ −13

2 3, , 2 12

1 2tt

t+−

≠, ,

15. 3 24 3

34

32

xx

x−+

≠ −, ,

16.a) b) c) 3:2x x

x

2 3 21

+ ++

x + 2

17. ( )( )xx

x++

=+1

6 11

6

3

2

18. a) b) c) d) 13 e) 440 f) 35n + 1 ( )( )n n+ +1 3 n + 3

19. 20. 21. 22. 23. x

x− 3

2u

u− 2

− 1 s ts t+−

xx−−

24

24. 25. 26. 1 21 2−+

yy

rr−−

14

− 1

27. 28. 29. 30. xx+−

11

1x y+

s ts t s t

2 2++ −( )( )

x xy yx y

2 2

2 2

+ ++

32

Multiplying or Dividing Rational Expressions

1. 2. 3. 23

02xy

aba b x y, , , , ≠

34

0x x y, , ≠ − ≠9 02 3x y x y, ,

4. 5. 6. 12

4, x ≠ y y−≠ −

22

1, − ≠ − ≠ab

a b a b6

0, , ,

7. 8. 9. xx

x+−

≠ − −21

6 3 1 5, , , , mm

m+≠ −

1 5 0 3 4, , , , xx y

x y y y+

≠ − ±6

6 3 7, , ,

10. 11. 4 52 3

13

32

aa

a−+

≠ ±, , ( )( )( )( )

, , , ,4 1 4 54 7 4 3

74

54

34

23

32

w ww w

w+ ++ +

≠ − − −

12. ( )( )( )

, , , , ,x xx

x+ −+

≠ − −5 2 1

2 11 3

40 1

312

13. 14. 3 12

2 1( ) , ,n n−≠ − ±

5 32 3

32

35

16

3 6xx

x++

≠ − −, , , , ,

15. a) b) c) d) no, the answer is c) is independent of x1027

2x 229

2x 335

16. 17. ( )( )x x− −3 2 3 2y +

18. a) b) c) 2( )( )6 9 2 42

x x− + ( )( )2 3 3 62

x x− +

19. xx−−

23 3( )

20. x 21. 22. 23. 24. 12t 3y r3

xx x( )( )− −2 1

25. 26. 27. 28. 29. 2 12

uu

++

xx+−

22

p qp q−+

uv u−

x yx y++ 3

Adding or Subtracting Rational Expressions

1. 2. 3. 20 2914

m + − −4 112y 3 1

2t +

4. a) b) 13 13x + 2 5 8, ,

5. a) b) c) ( )( )2 1 316

x x+ − ( )( )x x− −1 34

3 2 1 316

( )( )x x− −

33

d) e) 4 14

x − 3 2 1 316

( )( )x x− −

6. a) b) c) square d) n n( )+ 12

15 21 28 36 45, , , , ( )( )n n+ +1 22

e) f) It is a square.( )n + 1 2

7. 8. 9. 5 71 2

1 2xx x

x++ +

≠ − −( )( )

, , 8 31

0 1xx x

x−−

≠( )

, , 102 2

22x x

x xx−

− +≠ ±

( )( ),

10. 11. − +− +

≠ ±3 164 4

4yy y

y( )( )

, 9 23 4

3 42a a

a aa−

− −≠

( )( ), ,

12. 13. 18 92 1 3 1

13

12

nn n

n−− −

≠( )( )

, , − −+ −

≠ ±2 8

2 222

xx x

x( ) ( )

,

14. 15. 6 263 6 5

3 5 62m m

m m mm−

− − −≠

( )( )( ), , , 2 15

2 3 2 332

2

2

y yy y

y−− +

≠ ±( ) ( )

,

16. a) b) c) d) 6.75 h45s

452s

1352s

17. a) b) 2 12 1

0 12

xx

x+−

≠, , 4 12 2 1

12

yy

y+−

≠( )

,

c) d) mm

m+≠

12

0 1, , 22

2 02t

tt

+≠ −, ,

e) f) nn

n+−

≠ ±3

2 33

( ), 3 2 1

2 3 113

3( )( )

, ,xx

x++

≠ −

18. Answers may vary.

19. 20. 21. 22. 23. −14

37

t2

12

−+ −

21 12( ) ( )s s

24. 25. 52 2 3

xx x x( )( )( )− + −

42 1 2 12

tt t( ) ( )− +

26. 27. xx− 1 2

2 3 2

xy x y( )−

34

Rational Equations

1. 2. 12 3. 4. 5. 92

−32

2, 32

53

,− 23

32

,

6. 7. 8.75 h 8. 36 km13 13

h

9. No solution 10. 4 11. 2,3 12. 1,3 13. 0,3 14. !4

15. 20 h 16. 18 h 17. 2.5 km/h 18. 30