1 INTERMEDIATE ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE Directions

Level 3

1

INTERMEDIATE ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE

Directions: Study the examples, work the problems, then check your answers at the end of each topic. If you don't get the answer given, check your work and look for mistakes. If you have trouble, ask a math teacher or someone else who understands this topic.

TOPIC 1: ELEMENTARY OPERATIONS

A. Algebraic operations, grouping, evaluation: To evaluate an expression, first calculate the powers, then multiply and divide in order from left to right, and finally add and subtract in order from left to right. Parentheses have preference.

example: 14 - 32 = 14 - 9 = 5 example: 2 ? 4 + 3? 5 = 8 + 15 = 23 example: 10 - 2 ? 32 = 10 - 2 ? 9 = 10 -18 = -8 example: (10 - 2) ? 32 = 8 ? 9 = 72

Problems 1-7: Find the value:

1. 23 = 2. -24 =

3. 4 + 2 ? 5 = 4. 32 - 2 ? 3 + 1 =

5. 04 = 6. (-2)4 =

7. 15 =

Problems 8-13: Find the value if a = -3, b = 2 , c = 0, d = 1, and e = -3:

8. a - e =

11.

e d

+

b a

-

2d e

=

9. e2 + (d - ab)c =

12.

b e

=

10.

a - (bc - d) + e =

13.

d c

=

Combine like terms when possible:

example: 3x + y 2 - (x + 2y 2) = 3x - x + y2 - 2y2 = 2x - y2

example: a - a2 + a = 2a - a2

Problems 14-20: Simplify:

14. 6x + 3 - x - 7 = 18. 3a - 2(4(a - 2b) - 3a)=

15. 2(3 - t) =

19. 3(a + b) - 2(a - b) =

16. 10r - 5(2r - 3y) = 20. 1+ x - 2x + 3x - 4 x =

17. x2 - (x - x 2) =

B. Simplifying fractional expressions:

example:

27 36

=

9?3 9?4

=

9 9

?

3 4

= 1?

3 4

=

3 4

(note that you must be able to find a common

factor - in this case 9 - in both the top and

bottom in order to reduce a fraction.)

example:

3a 12ab

=

3a?1 3a?4 b

=

3a 3a

?

1 4b

= 1?

1 4b

=

1 4b

(common factor: 3a)

Problems 21-32: Reduce:

21.

13 52

=

22.

26 65

=

23.

3+ 6 3+ 9

=

24.

6axy 15by

=

25.

19a 2 95a

=

26.

14 x-7y 7y

=

27.

5a+b 5a+c

=

28.

x-4 4-x

=

29.

2( x+4 )(x-5) ( x-5)(x-4 )

=

30.

x 2 -9x x-9

=

31.

= 8( x -1)2

6( x 2 -1)

32.

= 2x 2 -x-1

x 2 -2x+1

example:

3 x

?

y 15

?

10x y2

=

3?10?x?y 15?x?y 2

=

3 3

?

5 5

?

2 1

?

x x

?

y y

?

1 y

=

1?1? 2 ?1?1?

1 y

=

2 y

Problems 33-34: Simplify:

33.

4x 6

?

xy y2

?

3y 2

=

34.

x 2 -3x x-4

?

x(x-4)

2x-6

=

C. Laws of integer exponents:

I. ab ? ac = ab+c

II.

ab ac

= ab-c

III. (ab )c = abc

IV. (ab)c = ac ? bc

( ) V.

= a c a c

b

bc

VI. a0 = 1 (if a 0 )

VII.

a-b

=

1 ab

Problems 35-44: Find x:

35. 23 ? 24 = 2x

36.

23 24

= 2x

37.

3-4

=

1 3x

38.

52 52

= 5x

( ) 39. 24 3 = 2x

40. 8 = 2x 41. ax = a3 ? a

42.

b 10 b5

=

bx

43.

1 c -4

= cx

44.

= a a 3y-2

x

a 2y-3

Problems 45-59: Simplify:

45. 8x0 = 46. 3-4 = 47. 23 ? 24 =

48. 05 = 49. 50 =

50. (-3)3 - 33 =

51. 2x ? 4x-1 =

52.

= 2 c + 3

2c-3

53. 2c+3 ? 2c-3 =

54.

= 8x

2 x -1

Copyright ? 1986, Ron Smith/Bishop Union High School, Bishop, CA 93514 Permission granted to copy for classroom use only. Not to be sold. One of a series of worksheets designed to provide remedial practice. Coordinated with topics on diagnostic tests supplied to the Mathematics Diagnostic Testing Project, Gayley Center Suite 304, UCLA, 405 Hilgard Ave., Los Angeles, CA 90024.

55.

= 2x -3

6x -4

( ) 56.

a x+3

x

=

57. = a 3x-2

a 2x-3

2

58. (-2a2 )4 (ab2 )=

( ) ( ) 59.

2 4 xy 2 -1 -2x-1y

2

=

D. Scientific notation:

example: 32800 = 3.2800 ?104 if the zeros in

the ten's and one's places are significant. If

the one's zero is not, write 3.280 ?104 ; if neither is significant: 3.28 ?104 example: .004031 = 4.031? 10-3 example: 2 ? 102 = 200 example: 9.9 ? 10-1 = .99

Note that scientific form always looks like a ?10n where 1 a < 10, and n is an integer power of 10.

Problems 60-63: Write in scientific notation:

60. 93,000,000 = 61. .000042 =

62. 5.07 = 63. -32 =

Problems 64-66: Write in standard notation:

64. 1.4030 ? 103 = 65. -9.11?10-2 =

66. 4 ? 10-6 =

To compute with numbers written in scientific form, separate the parts, compute, and then recombine.

( ) example: 3.14 ?105 (2) = (3.14) (2)?105

= 6.28 ?105

example:

4.28?106 2.14 ?10 -2

=

4.28 2.14

?

106 10 -2

= 2.00 ?108

Answers:

1. 8

2. ?16

3. 14

4. 4

5. 0

6. 16

7. 1

8. 0

9. 9

10. ?5

11. ?3 12. - 2 3

13. no value (undefined)

14. 5x - 4 15. 6 - 2t 16. 15y

17. 2x2 - x 18. a + 16b 19. a + 5b 20. 1 - 2x

21.

1 4

22.

2 5

23.

3 4

24.

2ax 5b

25.

a 5

26.

2x-y y

27.

5a +b 5a +c

28. -1

29.

2(x +4)

x-4

30. x

31.

4(x-1) 3(x +1)

32.

2x +1 x-1

33. x 2

34.

x2 2

35. 7

36. ?1

37. 4

example:

2.01?10 -3 8.04 ?10 -6

= .250 ?103 = 2.50 ?102

Problems 67-74: Write answer in scientific notation:

67. 1040 ?10-2 =

68.

= 10-40

10 -10

69.

1.86?104 3?10 -1

=

70.

3.6?10-5 1.8?10-8

=

71.

1.8?10-8 3.6?10-5

=

( ) 72. 4 ?10-3 2 =

( ) 73. 2.5 ?102 -1 =

74.

( )( ) = -2.92?103 4.1?107 -8.2?10-3

E. Absolute value:

example: 3 = 3 example: -3 = 3 example: a depends on a

if a 0, a = a if a < 0, a = -a example: - -3 = -3

Problems 75-78: Find the value:

75. 0 =

77. 3 + -3 =

76.

a a

=

78. 3 - -3 =

Problems 79-84: If x = -4 , find:

79. x + 1 = 80. 1- x = 81. - x =

82. x + x = 83. -3x = 84. (x - (x - x ) ) =

38. 0

39. 12

40. 3

41. 4

42. 5

43. 4 44. y + 1

45. 8

46.

1 81

47. 128

48. 0

49. 1

50. ?54

51. 2 3x-2

52. 64

53. 4c 54. 2 2x+1

55.

x 3

Copyright ? 1986, Ron Smith/Bishop Union High School, Bishop, CA 93514 Permission granted to copy for classroom use only. Not to be sold. One of a series of worksheets designed to provide remedial practice. Coordinated with topics on diagnostic tests supplied to the Mathematics Diagnostic Testing Project, Gayley Center Suite 304, UCLA, 405 Hilgard Ave., Los Angeles, CA 90024.

56. ax 2 + 3x 57. ax+1 58. 16a9b2

59.

2 x3

60. 9.3 ?107

61. 4.2 ?10-5

62. 5.07

63. ?3.2? 10

64. 1403.0

65. -.0911

3

66. .000004

67. 1? 1038 68. 1? 10-30 69. 6.2 ? 104 70. 2.0 ? 103 71. 5.0 ? 10-4 72. 1.6 ? 10-5 73. 4.0 ?10-3 74. 1.46 ? 1013

75. 0

76. 1 if a > 0; -1 if a < 0; (no value if a = 0 )

77. 6 78. 0 79. 3 80. 5 81. ?4 82. 0 83. 12 84. 12

TOPIC 2: RATIONAL EXPRESSIONS

A. Adding and subtracting fractions: If denominators are the same, combine the numerators:

example:

3x y

-

x y

=

3x - x y

=

2x y

Problems 1-5: Find the sum or difference as indicated (reduce if possible):

1.

4 7

+

2 7

=

2.

3 x-3

-

x x-3

=

3.

b-a b+a

-

a-b b+a

=

4.

- = x+2 3y 2

x 2 +2x xy 2

5.

3a b

+

2 b

-

a b

=

If denominators are different, find equivalent

fractions with common denominators:

example:

example: example:

34 is equivalent to how many eighths?

3 4

=

8;

3 4

= 1?

3 4

=

2 2

?

3 4

=

2?3 2?4

=

6 8

6 5a

=

5ab

;

6 5a

=

b b

?

6 5a

=

6b 5ab

3x + 2 x+1

=

4(x+1) ;

3x + 2 x+1

=

4 4

?

3x + 2 x+1

=

12 x + 8 4x+4

example:

x-1 x+1

=

(x+1)(x-2) ;

x-1 x+1

=

(x-2)(x-1) (x-2)(x+1)

=

x 2 -3x+2

(x+1)(x-2)

Problems 6-10: Complete:

6.

4 9

=

72

7.

3x 7

=

7y

8.

x+3 x+2

=

(x-1)(x+2)

9.

30-15a 15-15b

=

(1+ b )(1- b )

10.

x-6 6- x

=

-2

How to get the lowest common denominator (LCD) by finding the least common multiple (LCM) of all denominators:

example:

5 6

and

815 .

First find LCM of 6 and 15:

6 = 2?3

15 = 3? 5

LCM = 2 ? 3? 5 = 30

so,

5 6

=

25 30

,

and

8 15

=

16 30

example:

3 4

and

16a :

4 =2?2

6a = 2? 3? a

LCM = 2 ? 2 ? 3? a = 12a

so,

3 4

=

9a 12a

,

and

1 6a

=

2 12a

example:

2

3(x + 2)

and

ax 6( x+1)

3(x + 2) = 3? (x + 2)

6(x + 1) = 2 ? 3? (x + 1)

LCM = 2 ? 3? (x + 1)? (x + 2)

so,

2 3( x+2)

=

2?2(x+1) 2?3(x+1)(x+2)

=

4(x+1) 6(x+1)(x+2)

and

ax

6(x+1)

=

ax(x+2) 6(x+1)(x+2)

Problems 11-16: Find equivalent fractions with the lowest common denominator:

11.

2 3

and

2 9

12.

3 x

and

5

13.

x 3

and

-4 x+1

14.

3 x-2

and

4 2- x

15.

x

15(x 2 -2)

and

7x(y-1) 10(x-1)

16.

1 x

,

3x x+1

,

and

x2 x2+x

After finding equivalent fractions with common denominators, proceed as before (combine numerators):

example:

a 2

-

a 4

=

2a 4

-

a 4

=

2a-a 4

=

a 4

example:

3 x-1

+

1 x+2

=

3(x+2) (x-1)(x+2)

+

(x-1) (x-1)(x+2)

=

3x + 6+ x -1

(x-1)(x+2)

=

4x+5

(x-1)(x+2)

Problems 17-30: Find the sum or difference:

17.

3 a

-

1 2a

=

18.

3 x

-

2 a

=

19.

4 5

-

2 x

=

20.

2 5

+

2

=

21.

a b

-2=

22.

a-

c b

=

23.

1 a

+

1 b

=

24.

a-

1 a

=

25.

x x-1

+

x 1- x

=

26.

3x - 2 x-2

-

2 x+2

=

27.

2x-1 x+1

-

2x-1 x-2

=

4

28.

1

(x-1)(x-2)

+

1

(x - 2)(x - 3)

-

2

(x-3)(x-1)

=

29.

x x-2

-

4 x 2 -2x

=

30.

x x-2

-

4 x2-4

=

B. Multiplying fractions: Multiply the top numbers, multiply the bottom numbers, and reduce if possible.

example: example:

3 4

?

2 5

=

6 20

=

3 10

3(x+1)

x-2

?

x2-4 x 2 -1

=

3(x +1)(x + 2)(x - 2) (x - 2)(x +1)(x -1)

=

3x + 6 x-1

31.

2 3

?

3 8

=

32.

a b

?

c d

=

33.

2 7a

?

ab 12

=

34.

3(x + 4 )

5y

?

5y3 x 2 -16

=

35.

(a+b)3

( x-y )2

?

(x-y) (5- p)

?

( p-5)2 (a+b)2

=

( ) 36.

3 4

2=

( ) 37.

2a3 5b

3

=

( ) 38.

2

1 2

2=

C. Dividing fractions: Make a compound fraction and then multiply the top and bottom (of the big fraction) by the LCD of both:

example:

a b

?

c d

=

a b

c d

=

a b

c d

? bd ? bd

=

ad bc

( ) example:

2 3

7 -

1 2

=

2 3

7?6

-

1 2

?6

=

42 4-3

=

42 1

=

42

example:

5x 2y

?2x =

5x 2y

2x

=

5x 2y

?

2y

2x ?2y

=

5x 4 xy

=

5 4y

39.

3 4

?

2 3

=

40.

11

3 8

?

3 4

=

41.

3 4

?

2

=

42.

a b

?

3

=

43.

3 a

?

b 3

=

x+7

44.

x2-9 1

=

x-3

45.

a-4

3 a

-

2

=

46.

2a - b

1

=

2

47.

2

3

=

4

2

48. 3 = 4

a

49. b = c

50.

a

b

=

c

51.

1 a

1 a

- +

1 b

1 b

=

52.

1 2a

-

1 b

1 a

-

1 2b

=

53.

1 a

-

1 b

1

=

ab

Answers:

1.

6 7

2. ?1

3.

2b-2a b+a

4. - 2 x

5. 2a +2 b

6. 32

7. 3xy

8. x 2 + 2x - 3 9. 2 + 2b - a - ab

10. 2

11.

6 9

,

2 9

12.

3 x

,

5x x

13.

x(x 3(x

+1) +1)

,

-12

3(x +1)

14.

3 x-2

,

-4 x-2

( ) 15.

2x(x-1) , 21x(y-1) x 2 -2

( ) ( ) 30 x 2 -2 (x-1) 30 x 2 -2 (x-1)

16.

x +1

x(x +1)

,

, 3x 2

x(x +1)

x2

x(x +1)

17.

5 2a

18. 3a-2x ax

19.

4 x-10 5x

20.

12 5

21.

a-2b b

22.

ab - c b

23.

a+b ab

24. a 2 -1 a

25. 0

26.

3x2 +2x x2-4

27.

-3(2x-1) (x +1)(x-2)

28. 0

29.

x+2 x

30.

x2 +2x-4 x2-4

31.

1 4

32.

ac bd

33.

b 42

34.

3y 2 x-4

35.

(a +b)(5- p)

x-y

36.

9 16

37.

8a 9 125b 3

38.

25 4

39.

9 8

40.

91 6

41.

3 8

42.

a 3b

43. 9 ab

44.

x+7 x+3

45.

a2-4a 3- 2 a

46. 4a - 2b

47.

8 3

48.

1 6

49.

a bc

50.

ac b

51.

b-a b+a

52.

b-2a 2b-a

53. b - a

TOPIC 3: EXPONENTS and RADICALS

A. Definitions of powers and roots:

Problems 1-20: Find the value:

1. 23 = 2. 32 =

3. -42 =

11. 3 -125 = 12. 52 =

13. (-5)2 =

4. (-4)2 =

14. x 2 =

5. 04 = 6. 14 =

7. 64 = 8. 3 64 =

9. 6 64 = 10. - 49 =

15. 3 a3 =

16.

1 4

=

17. .04 =

( ) 18.

2 3

4

=

19. 4 81a8 =

20. 7 ? 7 =

B. Laws of integer exponents:

I.

ab ?ac = ab+c

II.

ab ac

= ab-c

( ) III. ab c = abc

IV. (ab)c = ac ? bc

( ) V.

= a c a c

b

bc

VI. a0 = 1 (if a 0 )

VII.

a-b

=

1 ab

Problems 21-30: Find x:

21. 23 ? 24 = 2x

22.

23 24

= 2x

23.

3-4

=

1 3x

24.

52 52

= 5x

( ) 25. 23 4 = 2x

26. 8 = 2x 27. a3 ? a = ax

28.

b 10 b5

= bx

29.

1 c -4

= cx

= a 30. a 3y-2 a 2y-3

x

Problems 31-43: Find the value:

31. 7x 0 = 32. 3-4 = 33. 23 ? 24 = 34. 05 =

35. 50 =

36. (-3)3 - 33 =

37. x c+3 ? x c-3 =

38. 2x ? 4 x-1 =

= 39. x c+3 x c-3

40.

8x 2 x -1

=

41.

2x -3 6x -4

=

( ) 42.

a x+3

x -3

=

43. = a 3x-2

a 2x-3

5

Problems 44-47: Write given two ways:

Given

44.

d -4 d4

( ) 45.

3x 3 -2 y

No negative No

powers

fraction

( ) 46.

a 2bc 3 2ab 2c

47. x 2y 3z -1

x 5y -6z -3

C. Laws of rational exponents, and radicals:

Assume all radicals are real numbers:

I. If r is a positive integer, p is an integer, and

( ) a 0, then

p

ar

=r

ap

=

r

a

p

which is a

real number. (Also true if r is a positive odd

integer and a < 0)

Think of

p r

as

power root

II.

r

ab

=r

a ?r

b, or

(ab) 1r

=

a

1 r

? b1r

( ) III.

r

a b

=

r r

a , or

b

= 1

ar

1

ar

b

1

br

IV. rs a = r s a = s r a

( ) ( ) 1

1

or

a1 rs

=

a1 s

= a r

1

r

s

Problems 48-53: Write as a radical:

48.

31 2

=

49.

42 3

=

( ) 50.

= 1

13 2

51.

x3 2

=

52. 2x 12 =

53. (2x)12 =

Problems 54-57: Write as a fractional power:

54. 5 = 55. 23 =

56. 3 a = 57. 1 =

a

Problems 58-62: Find x:

58. 4 ? 9 = x 59. x = 4

9

60. 3 64 = x

61. 3 64 = x

2

62.

x

=

83

3

42

Problems 63-64: Write with positive exponents:

( ) 63.

9x 6y-2

1 2

=

( ) 64. -8a6b-12 -23 =

D. Simplification of radicals:

example: 32 = 16 ? 2 = 4 2 example: 3 72 = 3 8 ? 3 9 = 23 9

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