Calculus Test questions

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
3+ 6
A. Algebraic operations, grouping, evaluation: 23. 3+ 9
= 28. x− 4
4 −x
=
To evaluate an expression, first calculate the
powers, then multiply and divide in order from 24.
6axy
15by
= 29.
2( x+4 )(x−5)
( x−5)(x−4 )
=
x 2 −9x
= =
2
left to right, and finally add and subtract in order 25. 19a
30.
95a x−9
from left to right. Parentheses have preference.
8( x−1) 2
26.
14 x−7y
= 31. =
14 − 3 = 14 − 9 = 5
2
example: 7y 6( x 2 −1)
example: 2 • 4 + 3• 5 = 8 + 15 = 23 27. 5a+b
= 32. 2x 2 −x−1
=
5a+c x 2 −2x+1
example: 10 − 2 • 32 = 10 − 2 • 9 = 10 −18 = −8
example: (10 − 2) • 32 = 8 • 9 = 72 example: 3
•y
x 15
• 10x
y2
=
3•10•x•y
15•x•y 2
=
Problems 1-7: Find the value: • • • • •1=
3 5 2 x
3 5 1 x y y
y


1. 23 = 5. 0 =
4
1 • 1 • 2 • 1 • 1 • 1y = 2y
2. −24 = 6. (− 2) =
4
Problems 33-34: Simplify:
3. 4 + 2•5 = 7. 1 =
5
x (x−4 )
x 2 −3x
4. 32 − 2 • 3 + 1 = 33. 4x
6
• yxy2 • 3y2 = 34. x−4
• 2x−6
=
Problems 8-13: Find the value if a = − 3 , b = 2 , C. Laws of integer exponents:
c = 0, d = 1, and e = − 3 :
I. a b • a c = a b+c
8. a − e = 11. e
d
+ − b
a
2d
e
= II. aa c = a b−c
b



9. e + (d − ab)c = =
2 b
12.
III. (a ) = a
b c bc
e
10. a − (bc − d) + e = 13. d
=
(ab)
c
c IV. = ac • bc
(ab )
Combine like terms when possible: c
V. = ac
bc
example: 3x + y 2 − (x + 2y 2 ) VI. a = 1 (if a ≠ 0 )
0

= 3x − x + y − 2y = 2x − y
2 2 2
VII. a− b = 1b
a
example: a − a + a = 2a − a
2 2

Problems 35-44: Find x:
Problems 14-20: Simplify:
35. 2 3 • 2 4 = 2 x 40. 8 = 2
x
14. 6x + 3 − x − 7 = 18. 3a − 2(4(a − 2b) − 3a ) =
36. 24 = 2 x 41. a x = a 3 • a
3

15. 2(3 − t) = 19. 3(a + b) − 2(a − b) = 2
16. 10r − 5(2r − 3y) = 20. 1+ x − 2x + 3x − 4 x = 37. 3−4 = 1 42. b 10 = bx
3x b5
17. x 2 − (x − x 2 ) =
38. 52 =5 x
43. 1 = cx
52 c −4

( ) =2
B. Simplifying fractional expressions: 4 3
= ax
x a 3 y− 2
9• 3 39. 2 44.
example: = 27
36
= • = 1• = 4
9• 4
9
9
3
4
3
4
3 a 2 y− 3
(note that you must be able to find a common Problems 45-59: Simplify:
factor - in this case 9 - in both the top and
45. 8x = 50. (−3) − 3 =
0 3 3
bottom in order to reduce a fraction.)
example: 123aba
= 33a•a•41b = 33aa • 41b = 1 • 41b = 41b 46. 3 =
−4
51. 2 x • 4 x −1 =
(common factor: 3a) 47. 2 3 • 2 4 = 52. 2 c−3 =
c+ 3

2
Problems 21-32: Reduce: 48. 0 = 53. 2 c +3 • 2 c −3 =
5


= = 49. 5 = 54. 8x−1 =
13 26 0 x
21. 52
22. 65 2

, 2
55. 2x −3

6 x −4
= (
58. − 2a ) (ab )=
2 4 2
example: 2.01×10 −3
8.04×10 −6
= .250 ×10 3 = 2.50 ×10 2
( )=x +3 x
2(4 xy ) (−2x y )
2 −1 −1 2
56. a 59. = Problems 67-74: Write answer in scientific notation:
−2
=
3 x− 2
57. a
67. 10 ×10 =
40
71. 1.8×10 −8 =
a 2 x− 3 3.6×10 −5

( )=
2
D. Scientific notation: 68. 10 −40 = 72. 4 ×10−3
10 −10
example: 32800 = 3.2800 ×10 if the zeros in
4

(2.5 ×10 ) =
−1
the ten’s and one’s places are significant. If 69. 1.86×10 4 = 73. 2
3×10 −1
the one’s zero is not, write 3.280 ×10 ; if
4
3.6×10 −5 (−2.92×10 )(4.1×10 )
3 7

70. = 74. =
neither is significant: 3.28 ×10
4
1.8×10 −8 −8.2×10 −3
−3
example: .004031 = 4.031× 10 E. Absolute value:
example: 2 × 10 = 200
2

−1 example: 3 = 3
example: 9.9 × 10 = .99
example: −3 = 3
Note that scientific form always looks like a ×10
n
example: a depends on a
where 1 ≤ a < 10 , and n is an integer power of 10.
if a ≥ 0 , a = a
Problems 60-63: Write in scientific notation:
if a < 0 , a = −a
60. 93,000,000 = 62. 5.07 = example: − −3 = −3
61. .000042 = 63. −32 =
Problems 75-78: Find the value:
Problems 64-66: Write in standard notation:
−6 75. 0 = 77. 3 + −3 =
64. 1.4030 × 10 = 66. 4 × 10 =
3
a
65. −9.11×10 =
−2
76. a
= 78. 3 − −3 =
To compute with numbers written in scientific form, Problems 79-84: If x = −4 , find:
separate the parts, compute, and then recombine.
79. x + 1 = 82. x + x =
example: ( )
3.14 ×10 5 (2) = (3.14 ) (2) ×10 5 80. 1− x = 83. −3 x =
= 6.28 ×10 5 81. − x = 84. ( x − ( x − x ) ) =
example: 4.28×10 6 = 4.28 × 10 6 = 2.00 ×10 8
−2
2.14×10 2.14 10 −2

Answers:
1. 8 21. 1 38. 0
4
2. –16 22. 2
5
39. 12
3. 14 23. 3 40. 3
4
4. 4 2ax 41. 4
24. 5b
5. 0 a
42. 5
6. 16 25. 5 43. 4
2x − y
7. 1 26. y 44. y +1
8. 0 5a +b 45. 8
27. 5a +c
9. 9 46. 1
28. -1 81
10. –5 2(x + 4 ) 47. 128
11. –3 29. x − 4
48. 0
12. − 2 3 30. x 49. 1
4 (x −1)
13. no value (undefined) 31. 3(x +1) 50. –54
14. 5 x − 4 32. 2x +1 51. 2 3x −2
15. 6 − 2t x−1
2 52. 64
16. 15 y 33. x2
34. x 53. 4c
17. 2 x − x 2 2x +1
2 2
54.
18. a + 16b 35. 7 55. x
3
19. a + 5b 36. –1
20. 1 − 2 x 37. 4

, 3
x 2 + 3x 66. .000004 76. 1 if a > 0;
56. a 38 -1 if a < 0;
x +1
57. a 67. 1× 10 (no value if a = 0 )
58. 16a b
9 2
68. 1× 10−30 77. 6
2
59. x 3 69. 6.2 × 104 78. 0
60. 9.3 ×10 7 70. 2.0 × 103 79. 3
61. 4.2 ×10−5 71. 5.0 × 10−4 80. 5
62. 5.07 72. 1.6 × 10−5 81. –4
63. –3.2 × 10 73. 4.0 ×10−3 82. 0
64. 1403.0 74. 1.46 × 1013 83. 12
65. -.0911 75. 0 84. 12

TOPIC 2: RATIONAL EXPRESSIONS

A. Adding and subtracting fractions: example: 3
4 and 1
6a :
If denominators are the same, combine the 4 = 2•2
numerators: 6a = 2 • 3• a
example: 3x
y
− xy = 3x − x
y
= 2x
y
LCM = 2 • 2 • 3• a = 12a
so, 43 = 12a
9a
, and 6a1 = 12a
2
Problems 1-5: Find the sum or difference as 2 ax
example: 3( x+2)
and 6( x +1)
indicated (reduce if possible):
x +2 3y 2 3(x + 2) = 3• (x + 2)
1. 4
+ 27 = 4. − =
x +2 x 6(x + 1) = 2 • 3• (x + 1)
2
7 xy 2
2. 3
x −3
− x −3
x
= 5. 3a
b
+ b2 − ab = LCM = 2 • 3• (x + 1) • (x + 2)
b−a
3. b+a
− ab −+ ba = so, 2
=
2•2(x +1) 4 (x +1)
= 6(x +1)(x +2) and
3( x +2) 2•3(x +1)(x +2)
ax (x +2)
If denominators are different, find equivalent ax
6(x +1)
= 6(x +1)(x +2)
fractions with common denominators:
example: 3 4 is equivalent to how many eighths? Problems 11-16: Find equivalent fractions with
3
= 8 ; 43 = 1• 43 = 22 • 43 = 2•4
2•3
= 68 the lowest common denominator:
4
2 2 3 4
example: 6
= 5 ab ; 56a = bb • 56a = 56ab b 11. 3
and 9
14. x −2
and 2− x
5a
3 x 7x (y−1)
= 4 (x+1) ; 3x+2 = 44 • 3x+2 = 12x+8 12. and 5 15. and 10(x−1)
15(x 2 −2)
3x+2
example: x+1 x+1 x+1 4 x+4
x
x −4 1 3x 2

example: x−1
= (x+1)(x−2) ; 13. 3
and x +1
16. , , and x x2 + x
x x +1
x+1
(x−2)(x−1) x 2 −3x+2 After finding equivalent fractions with common
x−1
x+1
= (x−2)(x+1) = (x+1 )(x−2) denominators, proceed as before (combine numerators):
2a− a
Problems 6-10: Complete: example: − a4 = 24a − a4 =
a
2 4
= a
4
6. 4
9
= 72 9. 30−15a
15−15b
= (1+b )(1−b ) 3
example: x −1 + x +2
1

3(x +2) (x −1)
7. 3x
7
= 7y 10. x −6
6− x
= −2 = (x −1)(x +2) + (x −1)(x +2)
8. x+3
= (x−1)(x+2) = (3xx−1+6+ x −1
= 4 x +5
x+2 )(x +2) (x −1)(x +2)
How to get the lowest common denominator Problems 17-30: Find the sum or difference:
(LCD) by finding the least common multiple
(LCM) of all denominators: 17. 3
a
− 21a = 23. 1
a
+ b1 =
example: 5 6 and 8 15 . 18. 3
x
− a2 = 24. a − a1 =
First find LCM of 6 and 15: 19. 4
− 2x = 25. x
x −1
+ 1−x x =
6 = 2• 3 5
3x −2
15 = 3• 5 20. 2
5
+2= 26. x −2
− x +2
2
=
2 x −1 2 x −1
LCM = 2 • 3• 5 = 30 21. a
b
−2= 27. x +1
− x −2 =
so, 6 = 25
5
30
8
, and 15 = 16
30 22. a − bc =