advanced algebra topics compass review
TRANSCRIPT
Advanced Algebra Topics COMPASS Review You will be allowed to use a calculator on the COMPASS test. Acceptable calculators are:
basic calculators, scientific calculators, and graphing calculators up through the level of the TI-86.
1. If 423)( 2 xxxf , find )2(f .
a. 20 b. 2 c. -8 d. 12
2. If 23)( xxf and 14)( 2 xxxg , find: ))(( xgf
a. 3 23 14 11 2 x x x b. 34 6 1 x x c. 2 1 x x d. 2 7 3 x x
3. If 23)( xxf and 14)( 2 xxxg , find: ))(( xgf
a. 2 7 3 x x b. 2 7 3 x x c. 2 1 x x d. 2 1 x x
4. If 23)( xxf and 14)( 2 xxxg , find: ))(( xfg
a. 23 8 1 x x b. 3 23 14 11 2 x x x c. 29 24 13 x x d. 2 1 x x
5. If 23)( xxf and 14)( 2 xxxg , find: )(xg
f
a. 2 4 1
3 2
x x
x b.
2
1
3x c.
1
2 d.
2
3 2
4 1
x
x x
6. If 23)( xxf and 14)( 2 xxxg , find: ))(( xgf
a. 3 2x b. 3 23 14 11 2 x x x c. 23 12 1 x x d. 29 24 13 x x
7. If 23)( xxf and 14)( 2 xxxg , find: ))(( xfg
a. 2 7 3 x x b. 2 4 1 x x c. 29 24 13 x x d. 23 12 1 x x
8. If 23)( xxf and 14)( 2 xxxg , find: ))2((gf 1
a. -12 b. -1 c. 1 d. -11
9. If 32)( xxf , find )(1 xf .
a. 1 3( )
2
xf x
b. 1( ) 2 3f x x c. 1 1( )
2 3f x
x
d. 1 1 1( )
2 3f x x
10. If 3 14)( xxf , find )(1 xf .
a.3
1 1( )
4
xf x
b. 31( ) 4 1f x x c. 1 3
1( ) 1
4f x x d. 1
3
1( )
4 1f x
x
11. If )(xf contains the point (4,1), then )(1 xf must contain what point?
a. (-4,1) b. (1,4) c. (4,-1) d. (1,1
4)
12. Find the domain of 2( ) 6 5f x x x
a. ( , ) b. [1,5] c. [3, ) d. [-4, )
13. Find the domain of 7
( )4 8
f xx
a. | 2x x b. | 0x x c. ( ,2) (2, ) d. ( ,0) (0, )
14. Find the domain of 82
1)(
2
xx
xxf
a. ( ,0) (0, ) b. ( ,1) (1, ) c. ( , 2) ( 2,1) (1,4) (4, ) d. ( , 2) ( 2,4) (4, )
15. Find the domain of ( ) 3 9 1f x x
a. [ 3, ) b. ( 3, ) c. (0, ) d. [0,3]
16. If 12)3(
)32( 2
x
x
kxx, find the value of k .
a. k = –7 b. k = – 5 c. k = 5 d. k = 7
17. If 14 yx and 53 xz , find an expression for z in terms of y .
a. z = 4y + 1 b. z = 12y – 2 c. z = 12y – 19 d. z = 3y – 5
18. Find the vertex of the function, 563)( 2 xxxf .
a. (-3,-5) b. (1,-8) c. (-1,-2) d. (-1,-8)
19. What are the zeros of the function, 382)( 2 xxxf ?
a. 2 2 10 b. 10
82
c. 8 2 10 d. 10
22
20. What are the zeros of the function, 1892)( 23 xxxxf ?
a. x = 3, x = –3, x = 2 c. x = 3, x = –3, x = – 2
b. x = 0, x = –3, x = 3, x = –2, x = 2 d. x = 0, x = 9, x = 2, x = – 2
21. Find the sum of the solutions of 452 2 xx .
a. 5
2 b. 5 c.
5
4 d. 4
22. Write a cubic function that has ,2,0 xx and 5x as zeros.
a. 3 2( ) 7 10f x x x x b. 3 2( ) 7 10f x x x x c. 3 2( ) 3 10f x x x x d. 3 2( ) 3 10f x x x x
23. Write a quadratic function that has a vertex at (2,-3) and contains the point (-2,5).
a. 2( ) ( 2) 3f x x b. 21( ) 2 1
2f x x x c. 2( ) ( 2) 3f x x d. 2( ) 4 1f x x x
24. Simplify 3
2
2 )(a
a. 8
3a b. 4
3a c. 2
3a d. 1
3a
25. Simplify 8
62
6
3
ab
ba
a. 22ab b. 2
2
ab c.
22
a
b d.
2
2
ab
26. Simplify
2
1
5
3
a
a
a. 1
10a b. 3
10a c. 1
5a d. 2
5a
27. Simplify 7
32 )4(x
x
a. 212x b. 1312x c.
64
x
d. 1364x
28. Simplify 2
5
2
3
2
3
2
1
23 baba
a. 3 15
4 46a b b. 2 45a b c. 3 86a b d. 2 46a b
29. Simplify
3
63
52
3
2
yx
yx
a. 3
3
2
3
y
x b.
3
3
8
27
y
x c.
33
15
27
8
y
x d.
15
33
3
2
x
y
30. Evaluate 2
1
6
1
88
a. 4 b. 1
1264 c. 1
128 d. 16
31. Evaluate 3
2
3
1
33
a. 1 b. 3 c. 2
99 d. Not possible
32. Simplify, assuming that the variables represent positive numbers:3 2aa
a. a a b. a c. 6a a d. Not possible
33. Simplify, assuming that the variables represent positive numbers: 53 ba
a. 15 ab b. 15 5 3a b c. 8 ab d. Not possible
34. Simplify, assuming that the variables represent positive numbers: baba 42 82
a. 34a b b. 34a b c. 44a b d. 6 216a b
35. Simplify, assuming that the variables represent positive numbers:4 354 52 44 baba
a. 7 816a b b. 42 3 2 32a b a b c. 3 44a b a d. 42 32ab a
36. Rewrite using logarithmic notation: 1642
a. 16log 4 2 b.
4log 2 16 c. 4log 16 2 d.
2log 16 4
37. Rewrite using logarithmic notation: yM x
a. logx y M b. log y M x c. logM x y d. logM y x
38. Rewrite using exponential notation: 2100log10
a. 210 100 b. 102 100 c. 2100 10 d. 1002 10
39. Rewrite using exponential notation: 13
1log3
a. 3 1( 1)
3 b. 1 1
33
c. 1
33 1 d. 3
11
3
40. Evaluate without a calculator:
4
1log2
a. 1
2
b. 2 c. -2 d.
1
16
41. Expand:3
2
logz
yxa
a. 2 3loga x y z b. 2log 3loga ax y z c. 2log log 3loga a ax y z d. 1
2log log log3
a a ax y z
42. Express as a single logarithm: tzy 10101010 log8log6log43log2
a. 106log (3 )yzt b.
10log (6 4 6 8 )y z t c. 4
10 6 8
9log
y
z t
d. 108log (3 )y z t
43. Simplify i2
to i, -i, 1, or -1.
a. i b. -i c. 1 d. -1
44. Simplify 8i to i, -i, 1, or -1.
a. i b. -i c. 1 d. -1
45. Simplify 11i to i, -i, 1, or -1.
a. i b. -i c. 1 d. -1
46. Add and write your answer in a+bi form: (1 + i) + (3 + 5i)
a. 4 + 6i b. -2 c. 4 – 6i d. 4 + 5i2
47. Subtract and write your answer in a+bi form:(3 – 4i) – (7 – i)
a. -4 – 5i b. 0 – 9i c. -4 – 3i d. 0 – 16i
48. Multiply and write your answer in a+bi form: 43 i
a. 0 + 12i b. 0 + 7i c. -12 d. 7
49. Multiply and write your answer in a+bi form: ii 52
a. 10i2 b. -10i c. -10 + 0i c. -10 + 0i
50. Multiply and write your answer in a+bi form: )52(7 ii
a. 35 + 14i b. 14 – 2i c. 2 – 35i d. -2 + 35i
51. Multiply and write your answer in a+bi form: )43)(21( ii
a. 3 + 8i b. 3 + 6i c. 5 – 10i d. -5 + 10i
52. Multiply and write your answer in a+bi form: 3)2( i
a. 8 – i3 b. 6 – i c. 2 – 11i d. 8 – 3i
53. Divide and write your answer in a+bi form: i3
2
a. 2
3 b. 6 + i c.
20
3i d. 6i
54. Divide and write your answer in a+bi form: i32
6
a. -1i b.
12 18
13 13i
c. 3 – 2i d. 12 – 18i
55. Divide and write your answer in a+bi form: 10 10
1 3
i
i
a. 2 + 4i b. -10 +
10
3i
c. 2 – 4i d. -40 – 2i
56. Solve for x over the complex number system: 0252 x
a. 5x i b. 5x c. 5x d. 5x
57. Solve for x over the complex number system: 422 xx
a. 1 3x i b. 2x c. 0,2x d. 2x i
58. Solve for x over the complex number system: 122 x
a. 3, 4x b. 2 3x i c. 3,4x d. 2 3x
59. Write the first five terms of the sequence having the general term 2
1 1)1(
n
na n
n
a. 3 4 5 6
2, , , ,4 9 16 25
b. 3 4 5 6
1, , , ,9 16 25 36
c. 2 3 4
0,1, , ,9 16 25
d. 1 3 4 5 6
, , , ,2 4 5 16 25
60. A certain arithmetic sequence has 561 a and 2611 a . Find 17a .
a. 21 b. 18 c. 3 d. 8
61. A certain geometric sequence has 641 a and 3245 a . Find 7a .
a. 2059 b. 454 c. 1358 d. 1358
62. Find the sum of the first ten terms of an arithmetic sequence having 271 a and 9d .
a. 117 b. 1.8x1010 c. 675 d. 108
63. Find the sum of the first ten terms of a geometric sequence having 30721 a and 2
3r .
a. 41,472 b. 6157.5 c. 348,150 d. 118,098
64. Evaluate
23
1
)36(n
n .
a. 1587 b. 60 c. 1518 d. 135
65. Evaluate
14
1 2
18
n
n
a. 15 b. 4 c. 1 d. 64
66. Evaluate 9!
3! 6!
a. 1 b. 84 c. 1
2 d.
45
126
67. Give the entry in the first row and the first column for
62
30
43
5
05
73
28
a. 11 b. 5 c. -7 d. 47
68. Give the entry in the second row and the first column for
30
52
43
65
32
a. 21 b. 33 c. 24 d. 26
69. Evaluate 53
912
a. 87 b. -33 c. -87 d. 33
70. Evaluate
132
065
213
a. 1 b. 0 c. 66 d. 7
71. Give the z-value of the solution to the following system:
12
72
1154
zy
zx
yx
a. 2 b. -1 c. 1 d. 4
72. If r
aS
1, then r = ?
a. S a
S
b. S a c. 1
S
a
d. 1S a
73. If 2 3 1 6 8
5 6 0 2 15 17
k
, then k =?
a. 3 b. 4 c. 8 d. 3
2
74. If is defined to be: 1 yxyx and 315x , then x = ?
a. 26 b. 6.2 c. 2 d. 36
75. If 1
4 2
k=10, then k = ?
a. 5
4 b. 3 c. 10 d. 7
76. In the list 12,5
3,2.16,
8
0,,25,7,
2
1 , the sum of all the rational numbers is:
a. 28.2 b. 33.3 c. 28.3 d. 5.5
77. How many integers are in the list: 12,5
3,2.16,
8
0,,25,7,
2
1 ?
a. 4 b. 3 c. 1 d. 6
78. Find the fourth term in the expansion of 7)3( zy .
a. 4 335y z b. 4 32835y z c. 4 31701y z d. 4 32835y z
79. How many terms are in the expansion of 7)3( zy ?
a. 2 b. 6 c. 7 d. 8
80. A set containing five elements has how many subsets?
a. 15 b. 31 c. 32 d. 10
Answers to Advanced Algebra Topics COMPASS Review
1. a
2. c
3. b
4. b
5. d
6. c
7. c
8. d
9. a
10. a
11. b
12. a
13. c
14. d
15. a
16. a
17. b
18. d
19. d
20. c
21. a
22. d
23. b
24. b
25. c
26. a
27. d
28. d
29. c
30. a
31. b
32. c
33. b
34. a
35. d
36. c
37. d
38. a
39. b
40. c
41. d
42. c
43. d
44. c
45. b
46. a
47. c
48. a
49. c
50. a
51. d
52. c
53. c
54. b
55. a
56. a
57. a
58. b
59. a
60. d
61. d
62. c
63. c
64. a
65. a
66. b
67. c
68. a
69. d
70. d
71. b
72. a
73. a
74. c
75. d
76. b
77. b
78. d
79. d
80. c
2/2012
Solutions to Advanced Algebra Topics COMPASS Review
1. If 423)( 2 xxxf , find )2(f .
f(-2) = 3(-2)2 – 2(-2) + 4 = 12 + 4 + 4 = 20
2. If 23)( xxf and 14)( 2 xxxg , find: ))(( xgf
(f + g) (x) = f(x) + g(x) = (3x – 2) + (x2 – 4x + 1) = x
2 – x – 1
3. If 23)( xxf and 14)( 2 xxxg , find: ))(( xgf
(f – g)(x) = f(x) – g(x) = (3x – 2) – (x2 – 4x + 1) = – x
2 + 7x – 3
4. If 23)( xxf and 14)( 2 xxxg , find: ))(( xfg
(f g)(x) = f(x) · g(x) = (3x – 2) (x2 – 4x + 1) = 3x
3 – 14x
2 + 11x – 2
5. If 23)( xxf and 14)( 2 xxxg , find: )(xg
f
)(xg
f
=
( )
( )
f x
g x =
2 4 1
3 2
x x
x
6. If 23)( xxf and 14)( 2 xxxg , find: ))(( xgf
))(( xgf = ( ( ))f g x = 2( 4 1) f x x = 23( 4 1) 2 x x = 23 12 1 x x
7. If 23)( xxf and 14)( 2 xxxg , find: ))(( xfg
))(( xfg = ( ( ))g f x = (3 2)g x = 2(3 2) 4(3 2) 1x x =
2(9 12 4) 12 8 1x x x =
29 24 13x x
8. If 23)( xxf and 14)( 2 xxxg , find: ))2((gf
))2((gf = ( 3)f = 3(–3) – 2 = – 9 – 2 = – 11
9. If 32)( xxf , find )(1 xf
32)( xxf Change the f(x) to y.
2 3y x Now reverse the roles of the x and y to form the inverse.
2 3x y This is the inverse, but now solve for y.
3
2
xy
Now name the function )(1 xf
)(1 xf = 3
2
x
10. If 3 14)( xxf , find )(1 xf .
3 14)( xxf
3 4 1y x Now reverse the roles of the x and y to form the inverse.
3 4 1x y This is the inverse, but now solve for y.
3
3 3 4 1x y
3 4 1x y
3 1
4
xy
)(1 xf =
3 1
4
x
11. If )(xf contains the point (4,1), then )(1 xf must contain what point?
The x and y values swap, so an ordered pair in )(1 xf is (1,4).
12. Find the domain of 2( ) 6 5f x x x
There will be restrictions on the domain if we have a square root (or any even root), if we have a variable in the
denominator of a fraction, or if we have a log. None of these occur in this function, so there are no restrictions on the
domain. The domain is ( , ) .
13. Find the domain of 7
( )4 8
f xx
The denominator cannot equal zero.
4 8 0x
2x
The domain is ( ,2) (2, ) .
14. Find the domain of 82
1)(
2
xx
xxf
The denominator cannot equal zero. 2 2 8 0x x
( 4)( 2) 0x x
4, 2x x
The domain is ( , 2) ( 2,4) (4, ) .
15. Find the domain of ( ) 3 9 1f x x
3 9x must be bigger than or equal to zero.
3 9 0x
3x
The domain is [ 3, ) .
16. If 12)3(
)32( 2
x
x
kxx, find the value of k .
Multiply by (x – 3) on both sides of the equation to get: 22 3 (2 1)( 3)x kx x x . 2 22 3 2 7 3x kx x x
So, k must be equal to -7.
17. If 14 yx and 53 xz , find an expression for z in terms of y .
Use substitution to get 3(4 1) 5z y .
12 2z y
18. Find the vertex of the function, 563)( 2 xxxf .
Method 1: Complete the square to obtain the standard form of 2( ) ( )f x a x h k , where the vertex is ( , )h k .
563)( 2 xxxf 2( ) 3( 2 ) 5f x x x 2( ) 3( 2 1) 5 3(1)f x x x
2( ) 3( 1) 8f x x
The vertex is ( 1, 8)
Method 2: x = 2
b
x
in 2( )f x ax bx c
563)( 2 xxxf
61
2(3)x
2( 1) 3( 1) 6( 1) 5f = – 8
The vertex is ( 1, 8) .
19. What are the zeros of the function, 382)( 2 xxxf ?
The zeros are the x-intercept values, where y = 0. 22 8 3y x x 20 2 8 3x x Since factoring is not possible, use the quadratic formula.
2( 8) ( 8) 4(2)(3)
2(2)x
4 1 10
2x
or
12 10
2x
20. What are the zeros of the function, 1892)( 23 xxxxf ?
This function is factorable by grouping: 2 2( ) ( 2) 9( 2) ( 9)( 2) ( 3)( 3)( 2)f x x x x x x x x x .
Set y = 0 to get zeros of 3, – 3 , and – 2.
21. Find the sum of the solutions of 452 2 xx .
The solutions of the equation are 5 57
4
.
5 57
4
+
5 57
4
=
5 5 5
4 4 2
22. Write a cubic function that has ,2,0 xx and 5x as zeros.
The zeros will create factors of ( )( 2)( 5)x x x .
Multiply these out to get 3 2( ) 3 10f x x x x .
23. Write a quadratic function that has a vertex at (2,-3) and contains the point (-2,5).
Use the standard form of a parabola: 2( ) ( )f x a x h k .
Insert the vertex: 2( ) ( 2) ( 3)f x a x .
Now use the other ordered pair: 25 ( 2 2) ( 3)a
1
2a
Now create the function using the a value: 21( ) ( 2) ( 3)
2f x x or f(x) = 21
2 12
x x
24. Simplify 3
2
2 )(a
Multiply the powers together.
25. Simplify 8
62
6
3
ab
ba
Take a factor of 3 out of the numerator and the denominator.
Subtract the powers of a to get 1a in the numerator.
Subtract the powers of b, starting in the denominator to get 2b in the denominator.
22
a
b
26. Simplify
2
1
5
3
a
a
Subtract the exponents: 3 1 6 5 1
5 2 10 10 10 . The result is
1
10a .
27. Simplify 7
32 )4(x
x
Take the power to a power first: 3 6
7
4 x
x.
Now deal with the negative exponent: 3 6 74 x x
Simplify: 1364x
28. Simplify 2
5
2
3
2
3
2
1
23 baba
Combine the constants, and combine the variables as appropriate. 1 3 3 5
2 2 2 26a b
= 2 46a b
29. Simplify
3
63
52
3
2
yx
yx
Deal with the outside power first, then the negative exponents, and then simplify. 3 6 15
3 9 18
2
3
x y
x y
=
3 15 18
3 6 9
3
2
y y
x x =
33
15
27
8
y
x
30. Evaluate 2
1
6
1
88
Add the exponents first: 1 1
6 28
= 2
38
Remember what the fractional exponent means and simplify: 2
3 8 = 22 = 4
31. Evaluate 3
2
3
1
33 1 2
3 33
= 3
32. Simplify, assuming that the variables represent positive numbers:3 2aa
21
32a a = 7
6a = 6 7a = 6a a
33. Simplify, assuming that the variables represent positive numbers: 53 ba 1 1
3 5a b = 5 3
15 15a b = 1
5 3 15( )a b = 15 5 3a b
34. Simplify, assuming that the variables represent positive numbers: baba 42 82
baba 42 82 = 6 216a b = 34a b
35. Simplify, assuming that the variables represent positive numbers:4 354 52 44 baba
2 5 5 34 4 4a a b b = 7 84 16a b = 2 342ab a
36. Rewrite using logarithmic notation: 1642
Remember the relationship between the exponential form of an equation and the logarithmic form:
logx
aa y y x to get 4log 16 2
37. Rewrite using logarithmic notation: yM x
Remember logx
aa y y x to get logM y x
38. Rewrite using exponential notation: 2100log10
Remember logx
aa y y x to get 210 100
39. Rewrite using exponential notation: 13
1log3
Remember logx
aa y y x to get 1 13
3
40. Evaluate without a calculator:
4
1log2
2
1log
4x
12
4
x 2
12
2
x 22 2x 2x
41. Expand:3
2
logz
yxa
Use the rules to expand a logarithm:
( ) log( ) log( )log ab a b
log( ) log( )a
log a bb
log logax a x
3
2
logz
yxa = 2 3log log loga a ax y z =
1
32log log loga a ax y z = 1
2log log log3
a a ax y z
42. Express as a single logarithm: tzy 10101010 log8log6log43log2
Use the rules above.
tzy 10101010 log8log6log43log2 = 2 4 6 8
10 10 10 10log 3 log log logy z t =
4 6 8
10 10 10 10log 9 log log logy z t = 4 6 8
10 10log (9 ) log ( )y z t = 4
10 6 8
9log
y
z t
43. Simplify i2
to i, -i, 1, or -1.
Remember that 2i is defined to be -1.
44. Simplify 8i to i, -i, 1, or -1.
Remember 4 2 2 1 1 1i i i
So, 8i = 4 4i i = 1 1 1
45. Simplify 11i to i, -i, 1, or -1. 11i = 4 4 2i i i i = (1)(1)(-1)i = -i
46. Add and write your answer in a+bi form: (1 + i) + (3 + 5i)
(1 + i) + (3 + 5i) = (1+3) + (i+ 5i) = 8 + 6i
47. Subtract and write your answer in a+bi form:(3 – 4i) – (7 – i)
(3 – 4i) – (7 – i) = 3 – 4i – 7 + i = -4 – 3i
48. Multiply and write your answer in a+bi form: 43 i
43 i = (3)(4)i = 12i
49. Multiply and write your answer in a+bi form: ii 52
ii 52 = 210i = (10)(-1) = - 10 and in a+bi form: -10 + 0i
50. Multiply and write your answer in a+bi form: )52(7 ii
)52(7 ii = 214 35i i = 14 35( 1)i = 35 14i
51. Multiply and write your answer in a+bi form: )43)(21( ii
)43)(21( ii = 3 10 8( 1)i = 5 10i
52. Multiply and write your answer in a+bi form: 3)2( i
(2 )(2 )(2 )i i i = 2(2 )(4 4 )i i i = (2 )(3 4 )i i = 2 11i
53. Divide and write your answer in a+bi form: i3
2
2
3
i
i i =
2
2
3
i
i =
2
3( 1)
i
=
2
3
i = 0
2
3i
54. Divide and write your answer in a+bi form: i32
6
6 2 3
2 3 2 3
i
i i
=
2
6(2 3 )
4 9
i
i
=
12 18
13
i =
12 18
13 13i
55. Divide and write your answer in a+bi form: 10 10
1 3
i
i
10 10 1 3
1 3 1 3
i i
i i
=
2
2
10 40 30
1 9
i i
i
=
20 40
10
i =
20 40
10 10i = 2 4i
56. Solve for x over the complex number system: 0252 x
0252 x 2 25x
2 25x
1 25x
5x i
57. Solve for x over the complex number system: 422 xx 2 2 4 0x x
Use the quadratic formula: 2( 2) ( 2) 4(1)(4)
2(1)
=
2 12
2
=
2 2 3
2 2i = 1 3 i or 1 3i
58. Solve for x over the complex number system: 122 x
122 x
2 12x
2 3x i
59. Write the first five terms of the sequence having the general term 2
1 1)1(
n
na n
n
1 1
1 2
1 1( 1)
1a =
21 2
1
2 1
2 2
2 1 3 3( 1) ( 1)
2 4 4a
3 1
3 2
3 1 4 4( 1) (1)
3 9 9a
4
5
16a and 5
6
25a
60. A certain arithmetic sequence has 561 a and 2611 a . Find 17a .
Use the formula for a particular term in an arithmetic sequence: 1 ( 1)( )na a n d
26 56 (11 1)( ) 26 56 10 3d d d
Since we now have 1a and d, use the formula to find
na : 56 ( 1)( 3) 56 3 3 59 3na n n n . Use this
formula to find 17a .
17 59 3(17) 8a
61. A certain geometric sequence has 641 a and 3245 a . Find 7a .
Use the formula for a particular term in a geometric sequence: 1
1
n
na a r
5 1 4 4324 64 5.0625 5.0625 1.5r r r r
Since we now have 1a and r, use the formula to find na :
13
642
n
na
6
7
364 729
2a
62. Find the sum of the first ten terms of an arithmetic sequence having 271 a and 9d .
Use the formula for the sum of an arithmetic sequence: 12 ( 1)2
n
nS a n d
10
102(27) (10 1)(9) 5 54 81 675
2S
63. Find the sum of the first ten terms of a geometric sequence having 30721 a and 2
3r .
Use the formula for the sum of a geometric sequence:
1
1
1
n
n
rS a
r
10
10
31
23072 348150
31
2
S
64. Evaluate
23
1
)36(n
n .
23
1
(6 3) (6 1 3) (6 2 3) (6 3 3) ... (6 23 3)n
n
= 3 + 9 + 15 + …+ 138.
We can see that the common difference is 6, so we will use the formula for the sum of an arithmetic series:
12 ( 1)2
n
nS a n d
23
232(3) (23 1)(6) 11.5 138 1587
2S
65. Evaluate
14
1 2
18
n
n
1 1 1 2 1 3 1 4 14
1
1 1 1 1 18 8 8 8 8 8 4 2 1 15
2 2 2 2 2
n
n
66. Evaluate 9!
3! 6!
9! 9 8 7 6! 9 8 784
3! 6! 3 2 1 6! 3 2 1
67. Give the entry in the first row and the first column for
62
30
43
5
05
73
28
Multiply each entry in the second matrix by 5. Add the first row first column entries together.
8 + -15 = -7
68. Give the entry in the second row and the first column for
30
52
43
65
32
Multiply the second row of the first matrix by the first column of the second to get this entry:
(– 2)(2) + (5)(5) = – 4 + 25 = 21
69. Evaluate 53
912
12 9(12)(5) ( 3)( 9) 60 27 33
3 5
70. Evaluate
132
065
213
Choose a row or column to work with, such as the third column.
3 1 25 6 3 1
5 6 0 (2) ( 1) (2)[ 15 ( 12)] ( 1)[ 18 ( 5)] (2)( 3) ( 1)( 13) 6 13 72 3 5 6
2 3 1
71. Give the z-value of the solution to the following system:
4 5 11
2 7
2 1
x y
x z
y z
Rewrite the system as:
4 5 11
2 7
2 1
x y
x z
y z
Multiply the 2nd
equation by -2 and add this new equation to the 1st equation to eliminate x .
Now we have two equations with two variables: 5 2 3
2 1
y z
y z
We want eliminate y now, so multiply the top equation by 2 and the bottom equation by 5. When we add them
together, we get z = -1.
72. If r
aS
1, then r = ?
(1 )1
a S aS S r a S rS a S a rS
r S
73. If 2 3 1 6 8
5 6 0 2 15 17
k
, then k =?
(2)(k) + (3)(0) = 6 and k = 3
74. If is defined to be: 1 yxyx and 315x , then x = ?
55 1x x
5 1 31x 5 32x
5 5 5 32x
2x
75. If 1
4 2
k=10, then k = ?
12 4 1 2 4
4 2
kk k
2k – 4 = 10
k = 7
76. In the list 12,5
3,2.16,
8
0,,25,7,
2
1 , the sum of all the rational numbers is:
1 0 3 1 325 16.2 12 5 0 16.2 12 33.3
2 8 5 2 5
77. How many integers are in the list: 12,5
3,2.16,
8
0,,25,7,
2
1 ?
025, , 12
8 are integers, since they are 5, 0, and 12. There are 3 integers.
78. Find the fourth term in the expansion of 7)3( zy .
Answer:
4 3 4 3 4 3
73 (35)(81) ( 1) 2835
3y z y z y z
79. How many terms are in the expansion of 7)3( zy ?
The number of terms is one more than the power, which makes it eight.
80. A set containing five elements has how many subsets?
There are 52 subsets.