please only do questions on the buttom page of 5657 ( R test only) make sure to show your work.
SECTION R.7
u.
\3/32
12.~
15.
17.
V72
efs4
16.
2 4
23.
2
–
4x
(vi – s) 2
(vs – \/6) 2
56. ( 1
+
V3) 2
s8. (V3 +
vz)
2
59. Surveying. A surveyor places poles at points A, B,
and C in order to measure the distance across a
pond. The distances AC and BC are measured as
shown. Find the distance AB across the pond.
20.
ef4ax9
v’x
57.
18.~
19. Vl28c d
21.
V48
V25o
14.
13.\/180
55.
51
Radical Notation and Rational Exponents
22. ~243m 5n10
+4
24. Yx 2 + 16x + 64
Simplify. Assume that no radicands were formed by
raising negative quantities to even powers.
25.
21.
29.
31.
V15 \/35
Vs\/10
26.
28.
V2l V6
v’uw
V2x3yVTuy
30. V3/z V20z
€x2y~ ~ – – 32. ~’¾x4y
+ 4)y’4(x + 4) 4
v14(x + 1) 2 y’1s(x + 1) 2
33. v12(x
34.
60. Distance fro m Airport. An airplane is flying at an.
altitude of 3700 ft. The slanted distance directly to
the airport is 14,200 ft. How far horizontally is the
airplane from the airport?
36.
35.
37• •
38. • 3 ~
39.–
40.
vsx
vsm
\/l28a2b4
–
V 16ab
42 .
43.
.
61. An equilateral triangle is shown below.
svz + 3\/32
46.
7Vl2 –
V45 + V80
48. 2 \/32 + 3 Vs – 4 Vl8
49. s\/2×2 – 6 V20x – sv’Bx2
so. 2€x2 + 5 ~ – 3Vx3
47. 6\/20 – 4
51.
52.
53.
54.
b
44
y
45.
y
cvs + 2Vs)(V8 –
2VS)
(V3 – \/2)(\.½ + \/2)
(2V3 + Ys)(V3 – 3Ys)
(\/6 – 4V7)(3V6 + 2V7)
2\/2
a
a
‘———y—‘———y—
a
a
2
2
a) Find an expression for its height h in terms of a.
b) Find an expression for its area A in terms of a.
62. An isosceles right triangle has legs of length s. Find
an expression for the length of the hypotenuse in
terms of s.
52
CHAPTER R
Basic Concepts of Algebra
63. The diagonal of a square has length 8 V2. Find the
length of a side of the square.
91. 125l/3
92.
64. The area of square PQRS is 100 ft2, and A, B, C, and
D are the midpoints of the sides. Find the area of
square ABCD.
93. as/4b3/4
94 . x2fsy1/s
95. ms/3n7/3
96. p7/6qll/6
p
Convert to exponential notation.
98. (efi3)s
97.
99. (
A
103.
‘Vs
ef4
2
1 –
• r,:
2v3 –
72.
1
V2
6
75 . • t
vm –
• r:
v6
.
r
vn
vso
77.–
6
3
V3
. r,:
V3
3 –
Va+ v’b
85.—
3a
3
r
.
113• (m1/2ns/2)2/3
114• (x5f3yif3z2/3)3/5
al
+ a4f3)
Write an expression containing a single radical and
simplify.
117.
V6V’2
121.
Wv?
va+
Vl2
5
124.
· ,rif!__
86.
~(x
vs
V2
~Vab
2
+
X
y)2efx+”y
Synthesis
V6
• r,:
s – v2
Simplify.
125.
+
VpVq
90. 4 7/ 2
120.
Vex + r) 3
3
1
viefs
123. – – – – – 4 ~ – – – – –
78.–
84.
118.
122.
V(a + x) V(a + x)
8 –
4b3/8
116. m2f3(m7/4 _ m5/4)
119.
t
+
. r
vq
Convert to radical notation and, if possible, simplify.
87. y 5/ 6
88. x 2 / 3
89. 16 3/ 4
112. a1//2bs/8
3
82.
VS
9
4
vv + vw
80
V11
+ vs
• r.:.
v2+3v7
3
79.fs
x2/3 )1/2
110. ( – – 2
4y
111.—–‘——
11 5 . a3f4( a2/3
• r,:
7 4. • r,:
76 . •
y12
xlf3yl/2
VS+
Rationalize the numerator.
83.
106.~
x2f3y5/6
70.fs
71. • r,:
v3 –
V’2
104.
Simplify and then, if appropriate, write radical notation.
107. (2a 312)(4a 112 )
108. (3a 516)(8a213)
68.
69.f!
81.
VS efs
109. ( 9;~4
v2s
73.
102.
66.J¾
ef7
ef2a2
R
Rationalize the denominator.
67 . • 3C::
100.
105.
D
65.J
V12)
4
101.
C
3z4/ 5
1
+
x2
x2
126.
Vl=7 – 2 Vl=7
1 – X
127.
(~)ya
128. (2a3b5/4cl/7)4 + (54a2b2/3c6/5)I/3
Summary and Review
53
Chapter R Summary and Review
STUDY GUIDE
I IMPORTANT PROPERTIES AND FORMULAS
I
Properties of the Real Numbers
Sum or Difference of Cubes
a+b=b+a;
ab= ba
Commutative:
a+ (b + c) = (a+ b) + c;
a(bc) = (ab)c
Associative:
Additive Identity:
a+0=0+a=a
Additive Inverse:
– a+a
Multiplicative Identity: a • 1
= a + (
a)
=o
= 1· a = a
1
a · – = – · a = 1 (a
a a
Distributive:
a(b + c) =ab+ ac
:;t:
O)
–
EquationSolving Principles
The Addition Principle: If a = b is true, then
a + c = b + cis true.
The Multiplication Principle: If a
The Principle of Square Roots: If x 2 = k, then x =
=
if a< 0.
vie.
Ifniseven,U=
Properties of Exponents
For any real numbers a and b and any integers m and n,
assuming 0 is not raised to a nonpositive power:
am• a"
= am+n
The Quotient Rule:
The Power Rule:
(am)"
Raising a Product to a Power: (ab
Raising a Quotient to a Power:
a)m
(b
am
= bm (b
:I
0)
vie or
Properties of Radicals
Let a and b be any real numbers or expressions for which the
given roots exist. For any natural numbers m and n ( n :I 1):
if a~ 0,
The Product Rule:
= b is true, then ac = be
The Principle of Zero Products: If ab = 0 is true, then a = 0
or b = 0, and ifa = 0 or b = 0, then ab = 0.
X
Absolute Value
For any real number a,
{a,a,
A3
is true.
1
Multiplicative Inverse:
lal =
= (A + B)(A2  AB + B2 )
2
B3 = (A  B)(A2 + AB + B )
A3 + B3
r
= amn
= ambm
If n is odd,
U
lal.
= a.
efa . v'b = efab.
n/a = efa
\/b
y'b (b :;c 0).
u
= (efar.
Pythagorean Theorem
LJ·
a2+b2=,2
a
Compound Interest Formula
A=
P(l + ;f
Special Products of Binomials
+ B) 2 = A2 + 2AB + B2
(A  B) 2 = A2  2AB + B2
(A + B)(A  B) = A2  B2
(A
Rational Exponents
For any real number a and any natural numbers m and n,
efa exists,
al/n = efa,
n :;t: 1, for which
am/n =
am/n
efam = (efat,
= _l_' a :;c
am/n
0.
and
54
CHAPTER R
Basic Concepts of Algebra
REVIEW EXERCISES
Answers for all the review exercises appear in the answer
section at the back of the book. If your answer is
incorrect, restudy the section indicated in red next to the
exercise or the direction line that precedes it.
Convert to decimal notation. [R.2]
17. 8.3 X 105
18. 2.07 X 10 7
Determine whether the statement is true or false.
I. If a < 0, then Ia I =  a. [R. 1]
2. For any real number a, a :f: 0, and any integers m
and n, am· an = amn. [R.2]
3. Ifa = bistrue,thena
+
c= b
+
cistrue. [R.5 ]
4. The domain of an algebraic expression is the set of all
real numbers for which the expression is defined. (R.6]
In Exercises 510, consider the following numbers
¾, \/17,
7, 43,
r,: 3
 V 2, 4
12
4, ?'
0, 2.191191119 ... ,
102,
'%4,
Convert to scientific notation. [R.2]
19. 405,000
20. 0.00000039
Compute. Write the answer using scientific
notation. [R.2]
21. (3.1 X 105 )( 4.5 X 103 )
2.5 X 108
22.
3.2 X 10
Simplify.
23. (3x 4y 5 )( 4x 2y) [R.2 ]
efs"5.
S. Which are rational numbers? [R.l ]
48a 3 b2 c5
24.
3 I 4
6a b c
6. Which are whole numbers? [R.I ]
\3/81
25.
7. Which are integers? [R.I ]
[R.7]
b 
9. Which are natural numbers? [R.I ]
::5
•
III
x2
y2
y
X
+
7}.
Simplify. [R. 1l
13
28.
2
y xy+x 2
V3  v0)( V3 + V7)
2
( 5  vz) [R.7]
30.
;:
31. 8 V:,
Calculate. [R.2]
32. (x
15. 3·2  4·2
4
+
(6  7) 4
16. __2_3___2_4_
3

6(3  1)
[R.6]
29. (
14. Find the distance between  5 and 5 on the number
line. [R.I ]
2
a 1
27. a  b1 [R.6)
IO. Which are irrational numbers? [R.I ]
12. 1241
[R.2]
[R.7]
26.
8. Which are real numbers? [R.I]
11. Write interval notation for { x I  4 < x
[R.I ]
13
33.
34.
25
+ VS
[R.7]
[R.7]
+ t) (x 2  xt + t 2 ) [R.3]
(Sa + 4b)(2a  3b) [R.3]
(6x 2y  3xy 2 + 5.xy  3) (4x 2y  4.xy 2 + 3.xy + 8) [R.3]
Summary and Review
V(a + b) 3 ~
59.
~(a+ b)7
Factor. [R.4)
35, 32X4  40xy3
3
+ 3y2  2y 
36. Y
37, 24x
55
+ 144 + X
7 5
•
60. Convert to radical notation: b l
6
2
[R.7]
61. Convert to exponential notation:
38, 9x3 + 35x2  4x
39. 9x2  30x + 25
40. 8x 3
41, 18X 2
62. Rationalize the denominator:
1

4
3x + 6

42. 4x3  4x
2

5
9x + 9
a2/J 
45,
zx 2 + 5x 
ab  6
64. Calculate: 128 + (2) 3 + (2)·3. [R.2]
3
8
A. 
3
46.zx7=7
= 3x  9
48. 8  3x =  7 + 2x
49. 6(2x  1) = 3  (x +
50. y2 + 16y + 64 = o
51. x 2  X = 20
52. 2x 2 + I lx  6 = o
53. x(x  2) = 3
54. y2  16 = 0
7
10)
Synthesis
Mortgage Paymen[ts
3x
2

12
+ 4x + 4
X
x
; x  2_ [R.6]
X
+2
+ 9x + 20
4
    . [R.6)
x 2 + 7x + 12
Write an expression containing a single radical. IR. 7]
58.
512
D. 3
96
2
•
[R.4]
l':;7 ]
M  p ( I
57. Subtract and simplify:
2
c.
=0
56. Divide and simplify:
x2
B. 24
65. Factor completely: 9x 2  36y
A. (3x + 6y)(3x  6y)
B. 3(x + 2y)(x  2y)
C. 9(x + 2y)(x  2y)
D. 9(x  2y) 2
47, 5x  7

[R.7]
of a 17ft pole to a point on the ground 8 ft from
the bottom of the pole? [R.7)
Solve. [R.5)
55. n2
yj
63. How long is a guy wire that reaches from the top
43, 6x 3 + 48
44,
+
V3
• r,:
v?efr2
+ ;,)' 
I
gives the monthly mortgage payment Mon a home loan
of P dollars at interest rater, where n is the total number
of payments (12 times the number of years). Use this
formula in Exercises 6~9. [R.:: 1
66. The cost of a house is $98,000. The down payment
is $16,000, the interest rate is 6½%, and the loan
period is 25 years. What is the monthly mortgage
payment?
56
Basic Concepts of Algebra
CHAPTER R
67. The cost of a house is $124,000. The down payment
is $20,000, the interest rate is 5¾%, and the loan
period is 30 years. What is the monthly mortgage
payment?
68. The cost of a house is $135,000. The down payment is
$18,000, the interest rate is 7½%, and the loan period
is 20 years. What is the monthly mortgage payment?
69. The cost of a house is $151,000. The down payment
is $21,000, the interest rate is 6¼%, and the loan
period is 25 years. What is the monthly mortgage
payment?
Multiply. Assume that all exponents are integers. [R. 3]
+ lO)(x"  4)
( t + t 0 )2
(l  zc)(l + zc)
70. (x"
71.
72.
0
+
75.
76.
m6n 
16y"
+
64
she probably making? [R.2]
78. When adding or subtracting rational expressions, we
can always find a common denominator by forming
the product of all the denominators. Explain why it
is usually preferable to find the least common
denominator. [R.6]
3xt  28
m3n
8. Convert to scientific notation: 4,509,000.
1. Consider the numbers
6 76 , •Vr.:::
12, 0,  13, .JI::
V 8, 1.2, 29, 5.
4
a)
b)
c)
d)
77. Anya says that 15  6 ; 3 • 4 is 12. What mistake is
80. Explain how you would determine whether
10V26  50 is positive or negative without
carrying out the actual computation. [R.7]
Factor. [R.4]
x 2t 
To the student and the instructor: The Collaborative
Discussion and Writing exercises are meant to be
answered with one or more sentences. These exercises
can also be discussed and answered collaboratively by
the entire class or by small groups. Answers to these
exercises appear at the back of the book.
79. Explain how the rule for factoring a sum of cubes can
be used to factor a difference of cubes. [R.4]
73. (a"  b") 3
74. y 2"
Collaborative Discussion and Writing
Which are whole numbers?
Which are irrational numbers?
Which are integers but not natural numbers?
Which are rational numbers but not integers?
3
I
•
Joi
4.
5. Write interval notation for { xi 3 <
graph the interval.
7. Calculate: 32 ; 2

answer:
2.7 X 104
3.6 X 103 ·
x
:5
12. (2y2)3(3y4)2
6}. Then
6. Find the distance between  9 and 6 on the number
line.
3
10. Compute and write scientific notation for the
Simplify.
11. x 8 ·x 5
Simplify.
2. l17.61
9. Convert to decimal notation: 8.6 X 10 5.
12 ; 4 • 3.
13. (3a 5 b4 )(5a 1b3 )
14. (sxy 4  7xy 2 + 4x 2  3)
(  3xy4 + 2xy 2  2y + 4)
15. (y  2)(3y
16. (4x  3)
2
+
4)
Test
32. Multiply and simplify:
x2 + x  6
~l_
y X
17, +
X y
18.\ffi
19, efs6
20. 3V75 + 2V27
21.
w vio
x2

4x
+ 4'
X
x2 
1
x2
+
3
4x  5 ·
34. Rationalize the denominator:
5
7 vT
35. Convert to radical notation: m 1 •
23, Bx 2
18

36. Convert to exponential notation:
24,
y2  3y 
25,
2n2
10x

8
27, m
18
+ Sn  12
+
3
2
+
25x
Solve.
29. 3(y  5)
2
38. Multiply: (x  y  1)2.
z2 
+
5x
11
+ 6 = 8  (y + 2)
+
=
0
3
=0
ef3"5.
37. How long is a guy wire that reaches from the top of a
12ft pole to a point on the ground 5 ft from the
bottom of the pole?
Synthesis
28. 7x  4 = 24
31.
x
25

38
Factor.
30. 2x
Bx+ 15
2
33. Subtract and simplify:
22. (2 + V3)(5  2\/3)
26. x 3
+
x2
57
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