Page 1

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Preface

This manual contains detailed solutions to all of the exercises of the text College Algebra, eleventh edition, by R. David Gustafson and Jeff Hughes. Many of the exercises in the text may be solved using more than one method, but it is not feasible to list all possible solutions in this manual. Also, some of the exercises may have been solved in this manual using a method that differs slightly from that presented in the text. There are a few exercises in the text whose solutions may vary from person to person. Some of these solutions may not have been included in this manual. For the solution to an exercise like this, the notation "answers may vary" has been included. If you are a student using this manual, please remember that only reading a solution does not teach you how to solve a problem. To repeat a commonly used phrase, mathematics is not a spectator sport. You MUST make an honest attempt to solve each exercise in the text without using this manual first. This manual should be viewed more or less as a last resort. Above all, DO NOT simply copy the solution from this manual onto your own paper. Doing so will not help you learn how to do the exercise, nor will it help you to do better on quizzes or tests. I would like to thank Paul McCombs from Rock Valley College and Cynthia Ashton of Brooks/Cole Publishing Company for their help and support. This solutions manual was prepared using EXP 5.1. This book is dedicated to John, who helps me to realize that mathematics cannot describe everything in life. May your study of this material be successful and rewarding. Michael G. Welden

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Contents

Chapter 0

Sets of Real Numbers

1

Chapter 1

Equations and Inequalities

52

Chapter 2

The Rectangular Coordinate System and Graphs of Equations

174

Chapter 3

Functions

236

Chapter 4

Exponential and Logarithmic Functions

309

Chapter 5

Solving Polynomial Equations

359

Chapter 6

Linear Systems

410

Chapter 7

Conic Sections and Quadratic Systems

505

Chapter 8

Sequence, Series and Probability

551

Chapter 9

The Mathematics of Finance

585

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SECTION 0.1

Exercises 0.1

(page 13)

1.

set

2.

subset

3.

union

4.

intersection

5.

decimal

6.

variable

7.

#

8.

even

9.

composite

10.

rational

11.

decimals

12.

Ÿ

13.

negative

14.

0

15.

B  C  D 

16.

CB

17.

&7  & † #

18.

commutative, multiplication

19.

interval

20.

no

21.

two

22.

half-open

23.

positive

24.

distance

25.

Every natural number is a whole number, so N § W. TRUE

26.

Every rational number is a real number, so Q § R. TRUE

27.

The rational number "# is not a natural ¸ N. FALSE number, so Q §

28.

Every integer is a rational number, so Z § Q. TRUE

29.

Every whole number is an integer, so W § Z. TRUE

30.

31.

E ∪ F œ ea, b, c, d, e, f, gf

32.

The real number È# is not an integer, ¸ Z. FALSE so R §

E ∩ F œ ed, ef

33.

E ∩ G œ ea, c, ef

34.

F ∪ G œ ea, c, d, e, f, gf

35.

* œ !Þ&'#&; terminates "'

36.

$ œ !Þ$(&; terminates )

37.

$ œ !Þ#(#(#(ÞÞÞ; repeats ""

38.

& œ !Þ%"''''ÞÞÞ; repeats "#

39.

natural: ", #, ', (

40.

whole: !, ", #, ', (

41.

integers: &, %, !, ", #, ', (

42.

43.

irrational: È#

rational: &, %,  #$ , !, ", #, #Þ(&, ', (

44.

prime: #, (

45.

composite: '

46.

even: %, !, #, '

47.

odd: &, ", (

48.

negative: &, %,  #$

49.

oqñqñqñqp # $ %

50.

oñqqñqqñqñp ' ) * %

1

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SECTION 0.1 51.

oñqñqqñqñp "" "$ "( "*

52.

oñqñqñqñqñqñqñp # " ! " # $ %

53.

oñqñqñqñp % $ # "

54.

oñqñqñqñp ) ' % #

55.

oñqñqñqñqñp & $ " " $

56.

oñqqqñqqqqñqp !Þ( "Þ(& $

57.

B  # p #, ∞

60.

58.

íïïÑqqp

#

%

61.

oqÐïïÑqp #

63.

$

# Ÿ B ! # p c#, #

64.

oq񕕄qp #

66.

B ! % p ∞, %

oqqÐïïî # ! B ! $ p #, $

67.

$

% ! B Ÿ " p %, "d

&

! 70.

$ Ÿ B ! % p c$, %

68.

oq񕕄qp

% Ÿ B Ÿ % p c%, %d

$

%

oqÒïïÓqp %

72.

$ € B € # p # Ÿ B Ÿ $ p c#, $d #

B  & and B ! %p &, ∞ ∩ ∞, %

oqÐïïïî & íïïïÑqp % oqÐïïÑqp &

%

oqÒïïÓqp

'

&, ∞ ∩ ∞, %

&

oqÐïïÓqp

' € B € # p # Ÿ B Ÿ ' p c#, 'd

∞, %

íïïÓqqp

& ! B Ÿ ! p &, !d

$

&, ∞

B Ÿ & p ∞, &d

65.

"

oqÒïïÓqp 73.

B ! $ p ∞, $

62.

íïïÑqqp

# Ÿ B Ÿ $ p c#, $d

#

&

%

oqÒïïÓqp 71.

!

oqqÐïïî

"

#

oqÐïïÑqp

B  % p %, ∞

%

oqqÒïïî 69.

! ! B ! & p !, &

59.

oqÐïïÓqp

#

B € " p c", ∞

( )

%

2

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$


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SECTION 0.1 74.

B € $ and B ! 'p c$, ∞ ∩ ∞, ' c$, ∞

oqÒïïïî $ íïïïÑqp ' oqÒïïÑqp

∞, '

c$, ∞ ∩ ∞, ' 75.

$

B € ) and B Ÿ $p c), ∞ ∩ ∞, $d c), ∞

∞, $d

c), ∞ ∩ ∞, $d 76.

oqÒïïïî ) íïïïÓqp $ oqÒïïÓqp )

B  " and B Ÿ (p ", ∞ ∩ ∞, (d ", ∞

∞, (d

", ∞ ∩ ∞, (d 77.

'

$

oqÐïïïî " íïïïÓqp ( oqÐïïÓqp "

(

B ! # or B  # p ∞, # ∪ #, ∞

78.

íïÑqqÐïî #

79.

íïÓqqÐïî

#

B Ÿ " or B € $ p ∞, "d ∪ c$, ∞

&

80.

íïÓqqÒïî "

81.

!

B ! $ or B € # p ∞, $ ∪ c#, ∞

íïÑqqÒïî

$

Since "$ € !, k"$k œ "$.

B Ÿ & or B  ! p ∞, &d ∪ !, ∞

$

82.

83.

Since ! € !, k!k œ !.

84.

85.

Since ) ! !, k)k œ ) œ ). k)k œ ) œ )

86.

#

Since "( ! !, k"(k œ "( œ "(. Since '$ € !, k'$k œ '$. k'$k œ '$ œ '$

Since #& ! !, k#&k œ #& œ #&.

3

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SECTION 0.1 87.

89.

91.

Since $# € !, k$#k œ $#. k$#k œ $# œ $#

88.

Since 1  & ! !, k1  &k œ 1  & œ 1  & œ &  1.

k1  1k œ k!k œ !

Since ' ! !, k'k œ ' œ '. k'k œ ' œ '

90.

Since )  1 € !, k)  1k œ )  1.

92.

Since #1 € !, k#1k œ #1.

93.

If B € #, then B  " € !. Then kB  "k œ B  ".

94.

If B Ÿ #, then B  " ! !. Then kB  "k œ B  ".

95.

If B ! !, then B  % ! !. Then kB  %k œ B  %.

96.

If B  "!, then B  ( € !. Then kB  (k œ B  (.

97. 99.

distance œ k)  $k œ k&k œ &

distance œ k$  )k œ k&k œ &

98.

distance œ k"#  &k œ k"(k œ "(

100. distance œ k#!  'k œ k#'k œ #'

101. Since population must be positive and never has a fractional part, the set of natural numbers should be used.

102. Since the subdivisions on a ruler are measured in fractions of an inch, the set of rational numbers should be used.

103. Since temperatures are usually reported without fractional parts and may be either positive or negative (or zero), the set of integers should be used.

104. Since the financial condition of a business is usually described in terms of dollars and cents (fractional parts of a dollar), the set of rational numbers should be used.

105. B will represent a positive number if B itself is negative. For instance, if B œ $, then B œ $ œ $, which is a positive number.

106. Every integer is a rational number because every integer is equal to itself over 1.

107. The statement is always true.

108. The statement is always true.

109. The statement is not always true. For example, let + œ & and , œ #.

110. The statement will be true if + € ! and , € !, or if + Ÿ ! and , Ÿ !.

111. The statement + ! ,  - could be interpreted to mean that +  -, when this is not necessarily true.

Exercises 0.2 1.

factor

112. k,  +k œ k"+  , k œ k"k † k+  , k œ k+  , k

(page 24) 2.

natural

3.

$, #B

4

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4.

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SECTION 0.2 7.

B7 B8 œ B78

8.

B7 8 œ B78

11.

B! œ "

12.

B8 œ

"$# œ "$ † "$ œ "'*

14.

"!$ œ "! † "! † "! œ ",!!!

15.

&# œ " † & † & œ #&

16.

&# œ && œ #&

17.

%B$ œ % † B † B † B

18.

19.

&B% œ &B&B&B&B

%B$ œ %B%B%B

20.

'B# œ ' † B † B

21.

)B% œ ) † B † B † B † B

22.

)B% œ )B)B)B)B

23.

(BBB œ (B$

24.

)CCCC œ )C%

25.

BB œ ""B# œ B#

26.

#+#+#+ œ # † # † # † +$ œ )+$

5.

scientific, integer

6.

9.

BC8 œ B8 C8

10.

13.

Answers may vary. B7 œ B78 B8

" B8

27.

$>$>$> œ $$$>$ œ #(>$

29.

BBBCC œ B$ C#

30.

+++,,,, œ +$ , %

31.

#Þ#$ œ "!Þ'%)

32.

(Þ"% œ #&%"Þ"')"

33.

!Þ&% œ !Þ!'#&

34.

!Þ#% œ !Þ!!"'

35.

B# B$ œ B#$ œ B&

36.

C$ C% œ C$% œ C(

37.

38.

>'  œ >'†( œ >%#

39.

C& C#  œ C(  œ C#"

40.

41. 43. 45. 47.

(

D #  D %  œ D ' D #! œ D #' &

$

#

$

51.

46.

B# C œ B#  C$ œ B' C$ $

+# +#  +' Œ  œ $ œ $ , , , $

49.

44.

$B$ œ $$ B$ œ #(B$ $

B! œ "

$

$

42.

+#  +%  œ +' +) œ +"%

D #  œ D #†$ œ D' $

+$ +' +% œ +* +% œ +"$

>$  >&  œ >"# >"! œ >## %

#

+#  +$  œ +) +* œ +"( %

$

#C% œ #% C% œ "'C%

48.

B$ D %  œ B$  D %  œ B") D #%

50.

Œ

$

52.

#, #, #, #,  œ " † # † # † # † # † , % œ "', %

28.

%B! œ % † " œ %

'

'

'

B B% B% œ œ  % C$ C"# C$  %

53.

5

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%B! œ "


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SECTION 0.2 54.

#B! œ # † " œ #

57.

C# C$ œ C& œ

59.

B$ B% 

61.

63.

D % œ

55.

" D%

" C&

58.

7# 7$ œ 7" œ 7

60.

C# C$ 

B( œ B($ œ B% B$

62.

<& œ <&# œ <$ <#

+#" œ +#""( œ +% +"(

64.

>"$ œ >"$% œ >* >%

66.

=* =$ ="# œ % œ ="#% œ =) # = =# 

68.

>% %$ $ " $ $ Œ $  œ ˆ> ‰ œ ˆ> ‰ œ > >

70.

<* <$ <' œ œ <'' œ <"# <' <# $

72.

Œ

74.

B$ B#  B"  B# " œ œ œ B"" œ "" $ $ * # & $ B B B B  B 

76.

B( C& C& C% C* C#( Œ ( %  œ Œ ( (  œ Œ "%  œ %# BC BB B B

#

œ B" 

#

œ B#

B#  B% œ œ B%$ œ B" œ B B# B B$ #

65.

67.

Œ

7$ 7$  7* œ œ  $ 8# 8' 8#  $

+$  ++#

#

69.

" œ ># >#

56.

œ

+$  , "

%

+' " œ +'$ œ +* œ * +$ +

Œ

73.

Œ

75.

B& C# B& B$ B) B$# Œ $ #  œ Œ # #  œ Œ %  œ "' B C C C C C

77.

&B$ C# Œ # $  $B C

œŒ

$B# C$ $B# B$ C# $B& *B"!  œŒ  œŒ  œ $ # $ &B C &C &C #&C#

78.

Œ

œŒ

#B# C' #C& # ) œ  Œ  œŒ %  œ # & # # ' $B C $B B C $B C #(B"# C$

79.

$B& C$ Œ & $  'B C

%

, " %

œ

œ

+"# ,%

<% <' <# # # % œ  Œ  œ ˆ< ‰ œ < $ $ ! < < < " œ % < #

#

%

%

#

$B# C&  #B# C'

$

#

" C%

>%  >$

>%  >) œ œ ># >' >$ #

71.

%

œ C% œ

$

$

+$ 

œ C" 

%

#

#

œ #

#

$

%

#

#

$

#

$

$

'B& C$ #C$ C$ #C' %C"# œ Œ & $  œ Œ & &  œ Œ "!  œ #! $B C "B B B B #

#

#

6

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$

$


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SECTION 0.2

80.

Œ

"#B% C$ D & $C$ C$ $C' #(C") œ œ œ  Œ  Œ  %B% C$ D & "B% B% D & D & B) D "! B#% D $!

81.

)# D $ C )# D $ C" '%D $ C" '%D $ D% '%D( œ $ ' ' " " # œ # & % œ œ $ " & " & C D †& C D & C D #&C C #&C' &C# D #  &CD # 

$

$

$

"

82.

7# 8$ :%  78# :$ 

83.



84.

'$  %  (# ‘ '$  $# ‘ 'c$  *d 'c'd $' ") œ œ œ œ œ # &#  %  &#  "' &"% (! (! $&

#

85. 87. 89. 91.

92. 93.

78# :$ % 78# :"

%

œ

7% 8' :) † 7% 8) :"# 7) 8"% :% 7) 7& :% :"$ 7"$ :"( œ & ' "$ œ œ % ) "# " # " ' "% 7 8 : †7 8 : 7 8 : 8 8 8#!

&c'#  *  &d &c$'  %d &c%!d #!! œ œ œ &! # # œ  %" % %#  $ %"

B# œ ## œ %

B$ œ #$ œ )

BD $ œ c" † # † $d$ œ '$ œ #"'

86. 88. 90.

B# œ ## œ " † % œ %

B$ œ #$ œ " † ) œ )

BD$ œ " † # † $$ œ # † #( œ &%

## † $$ ‘ B# D $  c% † #(d "!) œ œ œ œ "# # # # # D C $ ! *! *

$# ##  !# ‘ D # B#  C#  *%  ! *% $' $ œ œ œ œ œ $ B$ D ) $ #% #% #    # $

&B#  $C$ D œ &##  $!$ $ œ &%  $!$ œ #!  ! œ #!

94.

$B  D #  #C  D $ œ $#  $#  #!  $$ œ $&#  #$$ œ $#&  ##( œ (&  &% œ #"

95.

$B$ D # "D $ D $ $ $ $ œ œ & œ œ œ & œ 'B# D $ #B# B$ D # #B #  $#  '% '% ##

&B# D $  #&B% D ' &B% &B$ &#$ &) %! %! œ œ œ œ œ œ œ &BD # &BD # BD # D ' D% $% )" )" )" #

96. 97.

$(#,!!! œ $Þ(# ‚ "!&

98.

99.

"((,!!!,!!! œ "Þ(( ‚ "!)

100. #$,%(!,!!!,!!! œ #Þ$%( ‚ "!"!

101. !Þ!!( œ ( ‚ "!$

)*,&!! œ )Þ*& ‚ "!%

102. !Þ!!!&# œ &Þ# ‚ "!%

7

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SECTION 0.2 103. !Þ!!!!!!'*$ œ 'Þ*$ ‚ "!(

104. !Þ!!!!!!!)* œ )Þ* ‚ "!)

105. ",!!!,!!!,!!!,!!! œ " ‚ "!"#

106. !Þ!!!!!" œ " ‚ "!'

107. *Þ$( ‚ "!& œ *$(,!!!

108. %Þ#' ‚ "!* œ %,#'!,!!!,!!!

109. #Þ#" ‚ "!& œ !Þ!!!!##"

110. #Þ((% ‚ "!# œ !Þ!#((%

111. !Þ!!!$# ‚ "!% œ $Þ#

112. *,$!!Þ! ‚ "!% œ !Þ*$

113. $Þ# ‚ "!$ œ !Þ!!$#

114. (Þ#& ‚ "!$ œ (,#&!

115.

116.

117.

118.

'&,!!!%&,!!! 'Þ& ‚ "!% %Þ& ‚ "!%  'Þ&%Þ& œ œ ‚ "!%%& œ ""Þ( ‚ "!$ & #&!,!!! #Þ& ‚ "! #Þ& œ "Þ"( ‚ "!" ‚ "!$ œ "Þ"( ‚ "!% !Þ!!!!!!!%&!Þ!!!!!!"# %Þ& ‚ "!) "Þ# ‚ "!(  %Þ&"Þ# œ œ ‚ "!)(( ( %&,!!!,!!! %Þ& ‚ "! %Þ& œ "Þ# ‚ "!## !Þ!!!!!!$&"(!,!!! $Þ& ‚ "!( "Þ( ‚ "!&  $Þ&"Þ( œ œ ‚ "!(&( !Þ!!!!!!)& )Þ& ‚ "!( )Þ& œ !Þ( ‚ "!& œ ( ‚ "!" ‚ "!& œ ( ‚ "!% !Þ!!!!!!!"%%"#,!!! "Þ%% ‚ "!) "Þ# ‚ "!%  "Þ%%"Þ# œ œ ‚ "!)%& '!!,!!! ' ‚ "!& ' œ !Þ#)) ‚ "!* œ #Þ)) ‚ "!" ‚ "!* œ #Þ)) ‚ "!"!

119.

%&,!!!,!!!,!!!#"#,!!! %Þ& ‚ "!"! #Þ"# ‚ "!&  %Þ&#Þ"# œ œ ‚ "!"!&% !.!!!") "Þ) ‚ "!% "Þ) œ &Þ$ ‚ "!"*

120.

!Þ!!!!!!!!#(&%,(&! #Þ(& ‚ "!* %Þ(& ‚ "!$  #Þ(&%Þ(& œ œ ‚ "!*$"" "" &!!,!!!,!!!,!!! & ‚ "! & œ #Þ'"#& ‚ "!"(

121. $Þ$" ‚ "!% cm/sec œ

$Þ$" ‚ "!% cm "m '! sec $Þ$" ‚ "!% ' ‚ "!"  † † œ m/min " sec "!! cm " min " ‚ "!# $Þ$"' œ ‚ "!%"# m/min " œ "*Þ)' ‚ "!$ m/min œ "Þ*)' ‚ "!% m/min 8

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SECTION 0.2 122. Z œ 6A2 œ ',!!! mm*,(!! mm%,(!! mm œ ˆ' ‚ "!$ ‰ˆ*Þ( ‚ "!$ ‰ˆ%Þ( ‚ "!$ ‰ mm$ œ '*Þ(%Þ( ‚ "!$$$ mm$ œ #($Þ&% ‚ "!* mm$ œ #Þ($&% ‚ "!"" mm$

123. mass œ ",!!!,!!!,!!!!Þ!!!!!!!!!!!!!!!!!!!!!!!"'(#%) g œ ˆ" ‚ "!* ‰ˆ"Þ'(#%) ‚ "!#% g‰ œ "Þ'(#%) ‚ "!"& g $!,!!!,!!!,!!! cm " mile '! sec '! min † † † " sec "'!,*$%Þ% cm " min " hr $ ‚ "!"! ' ‚ "!" ' ‚ "!"  œ mile/hr "Þ'!*$%% ‚ "!& $'' œ ‚ "!"!""& mile/hr "Þ'!*$%% ¸ '(Þ"" ‚ "!( mile/hr œ 'Þ("" ‚ "!) mile/hr

124. $!,!!!,!!!,!!! cm/sec œ

125. Earth: 8 œ $ . œ *,#(&,#!!$ˆ#8# ‰  %‘ œ *,#(&,#!!$ˆ#$# ‰  %‘

œ *,#(&,#!!$ˆ#" ‰  %‘ œ *,#(&,#!!c"!d œ *#,(&#,!!! ¸ *$,!!!,!!! *Þ$ ‚ "!( mi

Mars: 8 œ % . œ *,#(&,#!!$ˆ#8# ‰  %‘ œ *,#(&,#!!$ˆ#%# ‰  %‘

œ *,#(&,#!!$ˆ## ‰  %‘ œ *,#(&,#!!c"'d œ "%),%!$,#!! ¸ "%),!!!,!!! "Þ%) ‚ "!) mi

126. "! † "! † "! † #' † #' † #' œ "!$ † #'$ ; "!$ † #'$ œ "(,&(',!!! œ "Þ(&(' ‚ "!( 127. polar radius œ 'Þ$&'(&!& ‚ "!$ km equatorial radius œ 'Þ$(8135 ‚ "!$ km

128. polar radius œ $Þ*%"")& ‚ "!$ mi equatorial radius œ $Þ*&%%%$( ‚ "!$ mi

129. B8 B# œ B8#

130.

B7 œ B7$ B$

132.

B$7& œ B$7&# œ B$7$ B#

131.

B7 B# B7# œ œ B7#$ œ B7" $ B B$

133. B7" B$ œ B7"$ œ B7%

134. +8$ +$ œ +8$$ œ +8

135. In the expression B% , the base of the exponent is B, while in the expression

136. $# ‚ "!# is not in scientific notation because $# is not a number between 1 and 10.

B% , the base of the exponent is B.

9

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SECTION 0.2 137. #, % Ê

138. ∞, $d ∪ c$, ∞ Ê

oqÐïïÑqp #

íïÓqqÒïî

%

$

139. Since 1  & ! !, k1  &k œ 1  & œ 1  & œ &  1.

Exercises 0.3

$

140. distance œ k&  (k œ k#k œ #

(page 38)

1.

0

2.

positive

3.

5.

+"Î8

6.

k+k

7.

9.

Á

11.

*"Î# œ $# 

13.

" Œ  #&

15.

)"

"Î%

17.

"!,!!!"Î% œ "!% 

19.

#( Œ  )

21.

10.

not

'# 

"Î$

4.

8 È +,

8 È + 8 È ,

8.

ÈB

)"Î$ œ #$ 

14.

"' Œ  '#&

16.

) Œ  #(

18.

",!#%"Î& œ %& 

œ%

20.

'%"Î$ œ ˆ%$ ‰

œ %

'%"Î# Ê not a real number

22.

23.

"'+# 

œ %+# ‘

"#&"Î$ œ &$ ‘

œ %k+k

24.

#&+% 

25.

"'+% 

œ ’#+% “

œ #k+k

26.

'%+$ 

27.

$#+& 

28.

'%+' 

29.

#"', ' 

30.

#&'>) 

œ$ # "Î#

" œ –Œ  — &

"Î#

œ $ 

% "Î%

"Î#

"Î&

œ "! $ "Î$

$ œ –Œ   — #

"Î#

œ #+& ‘

"Î$

"Î&

œ ’', #  “

œ

$ #

œ #+

$ "Î$

œ ', #

œ# % "Î%

# œ –Œ  — &

"Î%

" &

œ $

"Î%

"Î%

"Î$

"Î$

"Î%

"Î$

œ

#

78

12.

"Î#

, ˆ'"Î$ ‰

"Î$

"Î'

"Î$

œ ’&+#  “

œ %+$ ‘

œ #+' ‘

"Î%

"Î'

œ ’%>#  “

10

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œ

# $

œ & œ &k+# k œ &+#

# "Î#

"Î$

# &

$ "Î$

# œ –Œ  — $ "Î&

"Î#

œ

"Î$

œ %+

œ #k+k

% "Î%

œ %k># k œ %>#


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SECTION 0.3 "'+% Œ  #&, #

"Î#

31.

œ –Œ

# "Î#

%+# — &,

ϼ

"Î$

33.

37.

+&  $#, "!

34.

%*># Œ  "!!D %

"Î&

"Î#

œ ”Š 

+ & ‹ • #, #

"Î&

œ

# "Î#

(> œ –Œ  "!D # —

"!B# $C

ϼ

+ #, #

$

"'$Î# œ ˆ"'"Î# ‰ œ %$ œ '%

",!!!#Î$ œ ˆ",!!!"Î$ ‰ œ "!# œ "!!

41.

'%"Î# œ

43.

'%$Î# œ

45.

*$Î# œ 

36.

)#Î$ œ ˆ)"Î$ ‰ œ ## œ %

38.

)#Î$ œ ’)"Î$ “ œ ## œ %

#

#

$

39.

(> º "!D#

(k>k "!D #

œ

%$Î# œ ˆ%"Î# ‰ œ #$ œ )

40.

"!!$Î# œ ˆ"!!"Î# ‰ œ "!$ œ ",!!!

" " œ "Î# '% )

42.

#&"Î# œ

" " œ "Î# #& &

" " " " œ œ $ œ $ $Î# '% ) &"# '%"Î# 

44.

%*$Î# œ

" " " " œ œ $ œ $ $Î# %* ( $%$ %*"Î# 

46.

#(#Î$ œ

#

" " " œ $ œ  $ "Î# *$Î# $ *  " œ #( "Î# &

47.

% Œ  *

% œ –Œ  *

49.

#( Œ  '%

'% œ Œ  #(

50.

"#& Œ  )

) œŒ  "#&

51.

"!!=% 

œ "!!"Î# =% 

&Î#

Œ

$ "Î$

"!B# œ –Œ — $C œ

35.

32.

%+# &k, k

œ ",!!!B' Œ  #(C$

%+# º &,

#Î$

%Î$

"Î#

$

#& Œ  )"

&

%Î$

$Î#

48. "Î$ #

'% œ –Œ  #(

œ "!=#

œ

"

’#("Î$ “

#

" " # œ * $

"Î# $

#& œ –Œ  )"

& "#& — œ Œ *  œ (#* $

% "' — œ Œ $  œ * #

"Î$ %

) œ –Œ  "#&

"Î#

#(

#Î$

œ

# $# — œ Œ $  œ #%$ #Î$

"

# "' — œ Œ &  œ '#& %

52.

'%?' @$ 

"Î$

œ '%"Î$ ?' 

11

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"Î$

@$ 

"Î$

œ %?# @


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SECTION 0.3 53.

$#C"! D & 

54.

'#&+% , ) 

55.

B"! C& 

56.

"Î&

"Î%

$Î&

"

$#C"! D & "Î&

œ

"

"Î% '#&+% , ) 

œ

"

$#"Î& C"! "Î& D & "Î&

œ

"

"Î% '#&"Î% +%  , ) "Î%

œ

ˆ)B* C"# ‰#Î$ œ )#Î$ B")Î$ C#%Î$ œ

59.

Œ

60.

"'B% Œ  '#&C)

61.

#(<' Œ  ",!!!="#

62.

Œ

63.

+#Î& +%Î& +'Î& œ "Î& œ +&Î& œ + "Î& + +

)+'  "#&, *

œ

œ

" <' ="# "

)

#Î$

B' C) œ

)#Î$ +"#Î$ ## +% %+% œ œ #Î$ ")Î$ # ' "#& , & , #&, '

#Î$

$Î%

" " # ' ) œ %B' C) # B C

"'$Î% B"#Î% #$ B$ )B$ œ $ ' œ $Î% #%Î% '#& C & C "#&C'

#Î$

œŒ

$#7"!  #%$8"&

#Î&

",!!!="#  #(<'

œŒ

#Î$

œ

#%$8"&  $#7"!

#Î&

œ

",!!!#Î$ =#%Î$ "!# =) "!!=) œ # % œ #Î$ "#Î$ #( < $ < *<% #%$#Î& 8$!Î& $# 8' *8' œ œ $##Î& 7#!Î& # # 7% %7%

È%* œ È(# œ (

64. 66.

$ $ È "#& œ È &$ œ &

68.

$ $ È "#& œ É &$ œ &

70.

71.

Ê

73.

È$'B# œ É'B# œ k'Bk œ 'kBk

&

" &+, #

&

&Î'

58.

69.

œ

œ '%&Î' +$!Î' , '!Î' œ ˆ'%"Î' ‰ +& , "! œ #& +& , "! œ $#+& , "!

'%+' , "# 

ˆ<) ="' ‰$Î% œ <#%Î% =%)Î% œ <' ="# œ

67.

" #C# D

œ B$!Î& C"&Î& œ B' C$

57.

65.

œ

$# # # " & œ ËŒ  œ  œ "!!,!!! "! "! &

B'Î( B$Î( B*Î( œ (Î( œ B#Î( #Î( &Î( B B B

È)" œ È*# œ *

$ $ È '% œ É %$ œ %

& & È #%$ œ É $& œ $

72.

Ê

74.

È#&C# œ É&C# œ k&Ck œ &kCk

&

%

#&' % % % œ ËŒ  œ '#& & &

12

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%


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SECTION 0.3 75. 77.

79. 81. 82. 83. 84. 85.

86.

87. 88. 89. 90.

91.

92.

È*C% œ É$C# # œ k$C# k œ $C# $ $ È )C$ œ É #C$ œ #C

Ê %

76. 78.

B% C) BC# BC# kBkC# % œ œ œ Ë Œ  º º D "# D$ D$ kD $ k %

80.

È )  È # œ È %È #  È # œ #È #  È # œ È #

È+% , ) œ É+# , % # œ k+# , % k œ +# , % $ $ È #(D* œ É $D$ $ œ $D$

Ê &

+"! , & +# , +# , & œ œ Ë Œ  -"& -$ -$

È(&  #È#( œ È#&È$  #È*È$ œ &È$  #$È$ œ &È$  'È$ œ È$ È#!!B#  È*)B# œ È"!!B# È#  È%*B# È# œ "!BÈ#  (BÈ# œ "(BÈ#

È"#)+$  +È"'#+ œ È'%+# È#+  +È)"È#+ œ )+È#+  *+È#+ œ +È#+

#È%)C&  $CÈ"#C$ œ #È"'C% È$C  $CÈ%C# È$C œ #ˆ%C# ‰È$C  $C#CÈ $C œ )C# È$C  'C# È$C œ #C# È$C CÈ""#C  %È"(&C$ œ CÈ"'È(C  %È#&C# È(C œ C%È(C  %&CÈ(C œ %CÈ(C  #!CÈ(C œ #%CÈ(C

$ $ $ $ $ $ $ $ $ $ $ #È )"  $È #% œ #È #(È $  $È )È $ œ #$È $  $#È $ œ 'È $  'È $ œ "#È $ % % % % % % % % % % $È $#  #È "'# œ $È "'È #  #È )"È # œ $#È #  #$È # œ 'È #  'È #œ! % % % % % % % % % È (')D &  È %)D & œ È #&'D % È $D  È "'D% È $D œ %DÈ $D  #DÈ $D œ 'DÈ $D

& & & & & & & & #È '%C#  $È %)'C# œ #È $#È #C#  $È #%$È #C# œ ##È #C#  $$È #C# & & & œ %È#C#  *È#C# œ &È #C#

È)B# C  BÈ#C  È&!B# C œ È%B# È#C  BÈ#C  È#&B# È#C œ #BÈ#C  BÈ#C  &BÈ#C œ 'BÈ#C

$BÈ")B  #È#B$  È(#B$ œ $BÈ*È#B  #ÈB# È#B  È$'B# È#B œ $B$È#B  #BÈ#B  'BÈ#B œ *BÈ#B  #BÈ#B  'BÈ#B œ &BÈ#B

93.

&

$ $ $ $ $ $ $ $ È "'BC%  CÈ #BC  È &%BC% œ È )C$ È #BC  CÈ #BC  È #(C$ È #BC $ $ $ œ #CÈ#BC  CÈ#BC  $CÈ#BC œ !

13

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SECTION 0.3 94.

95.

97.

99.

101.

102.

103.

% % % % % % % % % È &"#B&  È$#B&  È ",#&!B& œ È#&'B% È #B  È"'B% È #B  È'#&B% È #B % % % % È È È È œ %B #B  #B #B  &B #B œ (B #B

$ $ È$ $È $ È œ † œ œ $ È$ È$ È$ $

' ' È& 'È & œ † œ È& È& È& &

96.

ÈB #È B # # œ † œ ÈB ÈB ÈB B

)È C ) ) ÈC œ † œ ÈC ÈC ÈC C

98.

$ $ $ # # È % #È % #È % È $ œ † œ œ œ % $ $ $ $ È È È È # ) # # %

$ $ $ %. %. È $ %. È $ %. È $ œ † œ œ $ $ $ $ È È È È $ * * $ #(

100.

$ $ $ È &+ &+ &+# &+È &+# &+È &+# $ œ † œ œ œ È&+# $ $ $ $ È#&+ È#&+ È&+# È"#&+$ &+

$ $ $ È ( ( '-# (È '-# (È '-# œ † œ œ $ $ $ $ È$'È$'- È'-# È#"'-$ '-

% % % È #, #, #(+# #, È #(+# #, È #(+# œ † œ œ % % % % È$+# È$+# È#(+# È)"+% $+

ÈB ÈB È#C È#BC B 104. Ê œ œ † œ È#C È#C È#C #C #C

$ $ $ $ $ $ È È #?% #?% ?$ È #? È $@# ?È '?@# ?È '?@# œ $ œ † œ œ $ $ $ È*@ È*@ È$@# È#(@$ *@ $@

105. Ê $

106. Ê $

107.

È&

109.

$ È *

110.

111.

"!

$

$ $ $ $ $ $ È È $=& $=& =$ È $=# È #< =È '<=# =È '<=# œ œ † œ œ  # $ $ $ $ È%<# È%<# È#< È)<$ %< #<

œ

œ

$ È "', #

"'

& È "', $

'%+

È& È& & " † œ œ "! È& "!È& #È &

108.

ÈC $

$ $ $ È È * È $ #( $ " † $ œ $ œ $ œ $ È$ $ È$ $È $ $È $

œ

œ

$ È "', #

"'

& È "', $

'%+

$ È %, $ È %,

& È #, #

È#, # &

œ

œ

$ È '%, $

$ "'È %,

œ

& È $#, &

& '%+È #, #

$ "'È %,

%,

œ

œ

œ

ÈC ÈC C † œ $ ÈC $È C

$ %È %,

,

#, , œ & È È # '%+ #, $#+ & #, # 14

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SECTION 0.3 112. Ê

È$B È$B È$B $B $B $B $B B œ œ † œ œ œ œ È&( È&( È$B È"("B È*È"*B È È &( $ "*B "*B

È" È" È$ È$ È$ È$ È$ " " " È$ " 113. Ê  Ê œ  œ †  † œ  œ  È$ È#( È$ È$ È#( È$ È)" $ #( $ $ * $È $ È $ #È $ œ  œ * * * $ $ $ $ $ $ $ $ È È È È È È " " " " È % " % % % % È % $ " $ œ $  $ œ $ † $  $ † $ œ $  $ œ  114. Ê  Ê È# È"' È# È% È"' È% È) È'% # "' # % $ $ $ È # % È % $È % œ  œ % % %

115. Ê

ÈB ÈB ÈB ÈB È# ÈB È# ÈB È# B B B Ê Ê œ   œ †  †  † È) È# È$# È) È# È# È# È$# È# ) # $# È#B È#B È#B œ   È"' È% È'% œ

È#B



È#B



È#B

% # ) È#B #È#B %È#B È#B œ   œ ) ) ) )

$ 116. Ê

$ $ $ $ C $ C $ C $ C $ C $ C È È È È È È È È È C C C # # # $ $ Ê Ê œ $  $  $ œ $ † $  $ † $  $ † $ È% È$# È&!! È% È# È$# È# È&!! È# % $# &!! $ $ $ È È #C È #C #C œ $  $  $ È) È'% È",!!!

œ

$ È #C



$ È #C



$ È #C

# % "! $ $ $ $ "!È #C &È #C #È #C "$È #C œ   œ #! #! #! #!

"Î% % 117. È * œ *"Î% œ $#  œ $#Î% œ $"Î# œ È$

"Î' ' 118. È #( œ #("Î' œ $$  œ $$Î' œ $"Î# œ È$

"Î"! "Î"! "Î& "! & 119. È "'B' œ "'B'  œ #% B'  œ #%Î"! B'Î"! œ ##Î& B$Î& œ ## B$  œ È %B$

"Î' "Î' "Î# ' 120. È #(B* œ #(B*  œ $$ B*  œ $$Î' B*Î' œ $"Î# B$Î# œ $B$  œ È$B$ œ BÈ$B ' ' ' $ ' ' 121. È#È # œ #"Î# † #"Î$ œ #$Î' † ##Î' œ È #$ È ## œ È )È %œÈ $#

' ' ' ' $ ' 122. È$È & œ $"Î# &"Î$ œ $$Î' &#Î' œ È $$ È &# œ È #(È #& œ È '(&

15

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SECTION 0.3

123.

% È $

124.

$ È #

È#

œ

È&

œ

% % % % % % È È È È È $"Î% $"Î% $ $ $ È % "# "# œ œ œ œ † œ œ % % % % % È È È È È # #"Î# ##Î% # "' % % % #

' ' ' ' ' ' È È È È È #"Î$ ##Î' ## % % È "#& &!! &!! œ œ œ œ † œ œ ' ' ' ' ' È È È È È $ &"Î# &$Î' & "&,'#& "#& "#& "#& &

% 125. ÈB% œ kBk. Since kBk œ B if B € !, then % È B% œ B if B € !.

B 127. Œ  C

7Î8

B B 8 126. Ê œŒ  C C

"Î8

œ

8 È B B"Î8 œ "Î8 8 È C C

B7Î8 B7Î8 B7Î8 C7Î8 C7Î8 C7 "Î8 C7 œ † œ œ œ Œ  C7Î8 C7Î8 B7Î8 C7Î8 B7Î8 B7 B7 "Î8

"Î8

œ

œÊ 8

C7 B7

128. Consider the case when 8 is even, 7 is odd and B is negative. Then 7 7 8 8 B7Î8 œ ˆB"Î8 ‰ œ ˆÈ B‰ . Thus, È B must be a real number for the expression to be defined. 129. # ! B Ÿ & Ê #, &d

131. B#  C# œ ##  $# œ %  * œ & 132.

130. If B  %, then $  B ! !. Then k$  Bk œ $  B œ $  B œ B  $.

BC  %C #$  %$ '  "# ' œ œ œ œ $ B # # #

133. '"(,!!!,!!! œ 'Þ"( ‚ "!)

Exercises 0.4

134. !Þ!!#$& ‚ "!% œ #$Þ&

(page 50)

1.

monomial, variables

2.

degree, variables

3.

trinomial

4.

binomial

5.

one

6.

zero

like

8.

degree

9.

coefficients, variables

10.

$È B  #

7.

12.

yes, binomial, 3rd degree

13.

15.

yes, binomial, 3rd degree

18.

no

21.

B$  $B#   &B$  )B œ B$  $B#  &B$  )B œ B$  &B$  $B#  )B œ 'B$  $B#  )B

11.

yes, trinomial, 2nd degree

no

14.

no

16.

yes, monomial, 5th degree

17.

yes, monomial, 0th degree

19.

yes, monomial, no degree

20.

yes, none, 3rd degree

16

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SECTION 0.4 22.

ˆ#B%  &B$ ‰  ˆ(B$  B%  #B‰ œ #B%  &B$  (B$  B%  #B

œ #B%  B%  &B$  (B$  #B œ B%  #B$  #B

23.

ˆC&  #C$  (‰  ˆC&  #C$  (‰ œ C&  #C$  (  C&  #C$  (

œ C&  C&  #C$  #C$  (  ( œ %C$  "%

24.

ˆ$>(  (>$  $‰  ˆ(>(  $>$  (‰ œ $>(  (>$  $  (>(  $>$  (

œ $>(  (>(  (>$  $>$  $  ( œ %>(  %>$  %

25.

#ˆB#  $B  "‰  $ˆB#  #B  %‰  % œ #ˆB# ‰  #$B  #"  $ˆB# ‰  $#B  $%  % œ #B#  'B  #  $B#  'B  "#  % œ #B#  $B#  'B  'B  #  "#  % œ B#  "%

26.

&ˆB$  )B  $‰  #ˆ$B#  &B‰  ( œ &ˆB$ ‰  &)B  &$  #ˆ$B# ‰  #&B  (

œ &B$  %!B  "&  'B#  "!B  ( œ &B$  'B#  %!B  "!B  "&  ( œ &B$  'B#  $!B  )

27.

)ˆ>#  #>  &‰  %ˆ>#  $>  #‰  'ˆ#>#  )‰ œ )ˆ># ‰  )#>  )&  %ˆ># ‰  %$>  %#  'ˆ#># ‰  ') œ )>#  "'>  %!  %>#  "#>  )  "#>#  %) œ )>#  %>#  "#>#  "'>  "#>  %!  )  %) œ #)>  *'

28.

$ˆB$  B‰  #ˆB#  B‰  $ˆB$  #B‰ œ $ˆB$ ‰  $B  #ˆB# ‰  #B  $ˆB$ ‰  $#B œ $B$  $B  #B#  #B  $B$  'B œ $B$  $B$  #B#  $B  #B  'B œ #B#  B

29.

CˆC#  "‰  C# C  #  C#C  # œ CˆC# ‰  C"  C# C  C# #  C#C  C# œ C$  C  C$  #C#  #C#  #C œ C$  C$  #C#  #C#  C  #C œ %C#  C

30.

%+# +  "  $+ˆ+#  %‰  +# +  #

œ %+# +  %+# "  $+ˆ+# ‰  $+%  +# +  +# #

œ %+$  %+#  $+$  "#+  +$  #+# œ %+$  $+$  +$  %+#  #+#  "#+ œ #+$  '+#  "#+

31.

BCB  %C  CˆB#  $BC‰  BC#B  $C œ BCB  BC%C  CˆB# ‰  C$BC  BC#B  BC$C

œ B# C  %BC#  B# C  $BC#  #B# C  $BC# œ B# C  B# C  #B# C  %BC#  $BC#  $BC# œ #B# C  %BC#

17

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SECTION 0.4 32.

33.

$787  #8  '7$78  "  #8%78  " œ $787  $78#8  '7$78  '7"  #8%78  #8" œ $7# 8  '78#  ")7# 8  '7  )78#  #8 œ $7# 8  ")7# 8  '78#  )78#  '7  #8 œ "&7# 8  #78#  '7  #8 #B# C$ %BC%  œ #%B# BC$ C% œ )B$ C(

34.

"&+$ , ˆ#+# , $ ‰ œ "&#+$ +# ,, $ œ $!+& , %

35.

$7# 8ˆ#78# ‰Š

36.



37.

%<=<#  =#  œ %<=<#   %<==#  œ %<$ =  %<=$

38. 39.

78 " ' % % 7% 8% 7 8 œ ‹ œ $#Œ 7# 7788# 8 œ "# "# "# #

$<# =$ #<# = "&<=# $ # "& # # $ # & ' Œ Œ  œ Œ Œ Œ < < <= == œ $< = & $ # & $ #

'?# @#?@#  C œ '?# @#?@#   '?# @C œ "#?$ @$  '?# @C

'+, # -ˆ#+-  $,-#  %+, # -‰ œ '+, # -#+-  '+, # -ˆ$,-# ‰  '+, # -ˆ%+, # -‰ œ "#+# , # -#  ")+, $ -$  #%+# , % -#

78# 78# 78# 78# ˆ%78  '7#  )‰ œ  ˆ'7# ‰  %78  ) # # # # # $ $ # # œ #7 8  $7 8  %78

40.



41.

+  #+  # œ +#  #+  #+  % œ +#  %+  %

43.

+  '# œ +  '+  ' œ +#  '+  '+  $' œ +#  "#+  $'

42.

44.

C  &C  & œ C#  &C  &C  #& œ C#  "!C  #& >  *# œ >  *>  * œ >#  *>  *>  )" œ >#  ")>  )"

45.

B  %B  % œ B#  %B  %B  "' œ B#  "'

46.

D  (D  ( œ D #  (D  (D  %* œ D #  %*

47.

B  $B  & œ B#  &B  $B  "& œ B#  #B  "&

48.

D  %D  ' œ D #  'D  %D  #% œ D #  #D  #%

49.

?  #$?  # œ $?#  #?  '?  % œ $?#  %?  %

50.

%B  "#B  $ œ )B#  "#B  #B  $ œ )B#  "!B  $

51.

&B  "#B  $ œ "!B#  "&B  #B  $ œ "!B#  "$B  $

52.

%B  "#B  ( œ )B#  #)B  #B  ( œ )B#  $!B  (

18

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SECTION 0.4 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

$+  #, # œ $+  #, $+  #,  œ *+#  '+,  '+,  %, # œ *+#  "#+,  %, #

%+  &, %+  &,  œ "'+#  #!+,  #!+,  #&, # œ "'+#  #&, #

$7  %8$7  %8 œ *7#  "#78  "#78  "'8# œ *7#  "'8#

%<  $=# œ %<  $=%<  $= œ "'<#  "#<=  "#<=  *=# œ "'<#  #%<=  *=# #C  %B$C  #B œ 'C#  %BC  "#BC  )B# œ 'C#  "'BC  )B#

#B  $C$B  C œ 'B#  #BC  *BC  $C# œ 'B#  (BC  $C#

*B  CB#  $C œ *B$  #(BC  B# C  $C# œ *B$  B# C  #(BC  $C# )+#  , +  #,  œ )+$  "'+# ,  +,  #, #

&D  #>D #  > œ &D$  &>D  #>D#  #># œ &D$  #>D#  &>D  #>#

C  #B# B#  $C œ B# C  $C#  #B%  'B# C œ #B%  &B# C  $C#

ŠÈ&  $B‹Š#  È&B‹ œ #È&  &B  'B  $È&B# œ $È& B#  B  #È& ŠÈ#  B‹Š$  È#B‹ œ $È#  #B  $B  È#B# œ È# B#  &B  $È# $B  "$ œ $B  "$B  "$B  " œ ˆ*B#  $B  $B  "‰$B  " œ ˆ*B#  'B  "‰$B  "

œ *B# $B  *B# "  'B$B  'B"  "$B  "" œ #(B$  *B#  ")B#  'B  $B  " œ #(B$  #(B#  *B  "

66.

#B  $$ œ #B  $#B  $#B  $ œ ˆ%B#  'B  'B  *‰#B  $ œ ˆ%B#  "#B  *‰#B  $

œ %B# #B  %B# $  "#B#B  "#B$  *#B  *$ œ )B$  "#B#  #%B#  $'B  ")B  #( œ )B$  $'B#  &%B  #(

67.

$B  "ˆ#B#  %B  $‰ œ $Bˆ#B# ‰  $B%B  $B$  "ˆ#B# ‰  "%B  "$

œ 'B$  "#B#  *B  #B#  %B  $ œ 'B$  "%B#  &B  $

68.

#B  &ˆB#  $B  #‰ œ #BˆB# ‰  #B$B  #B#  &ˆB# ‰  &$B  &#

œ #B$  'B#  %B  &B#  "&B  "! œ #B$  ""B#  "*B  "!

19

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SECTION 0.4 69.

$B  #Cˆ#B#  $BC  %C# ‰ œ $Bˆ#B# ‰  $B$BC  $Bˆ%C# ‰  #Cˆ#B# ‰  #C$BC  #Cˆ%C# ‰

œ 'B$  *B# C  "#BC#  %B# C  'BC#  )C$ œ 'B$  &B# C  'BC#  )C$

70.

%<  $=ˆ#<#  %<=  #=# ‰ œ %<ˆ#<# ‰  %<%<=  %<ˆ#=# ‰  $=ˆ#<# ‰  $=%<=  $=ˆ#=# ‰

œ )<$  "'<# =  )<=#  '<# =  "#<=#  '=$ œ )<$  "!<# =  #!<=#  '=$

71. 72.

#C8 $C8  C8  œ #C8 $C8   #C8 C8  œ 'C88  #C88 œ 'C#8  #C! œ 'C#8  # $+8 ˆ#+8  $+8" ‰ œ $+8 #+8   $+8 ˆ$+8" ‰ œ '+88  *+88" œ '+!  *+" * œ' +

73.

&B#8 C8 ˆ#B#8 C8  $B#8 C8 ‰ œ &B#8 C8 ˆ#B#8 C8 ‰  &B#8 C8 ˆ$B#8 C8 ‰

74.

#+$8 , #8 ˆ&+$8 ,  +, #8 ‰ œ #+$8 , #8 ˆ&+$8 , ‰  #+$8 , #8 ˆ+, #8 ‰

75.

B8  $B8  % œ B8 B8  %B8  $B8  "# œ B#8  B8  "#

76.

œ "!B#8#8 C88  "&B#8#8 C88 œ "!B%8 C!  "&B! C#8 œ "!B%8  "&C#8

œ "!+$8$8 , #8"  #+$8" , #8#8 œ "!+! , #8"  #+$8" , ! œ "!, #8"  #+$8"

+8  &+8  $ œ +8 +8  $+8  &+8  "& œ +#8  )+8  "&

77.

#<8  ($<8  # œ #<8 $<8   #<8 #  ($<8   "% œ '<#8  %<8  #"<8  "% œ '<#8  #&<8  "%

78.

%D 8  $$D 8  " œ %D8 $D8   %D8 "  $$D8   $ œ "#D #8  %D 8  *D8  $ œ "#D#8  "$D8  $

79. 80. 81.

B"Î# ˆB"Î# C  BC"Î# ‰ œ B"Î# B"Î# C  B"Î# BC"Î# œ B#Î# C  B$Î# C"Î# œ BC  B$Î# C"Î#

+, "Î# ˆ+"Î# , "Î#  , "Î# ‰ œ +, "Î# +"Î# , "Î#  +, "Î# , "Î# œ +$Î# , #Î#  +, #Î# œ +$Î# ,  +, ˆ+"Î#  , "Î# ‰ˆ+"Î#  , "Î# ‰ œ +"Î# +"Î#  +"Î# , "Î#  +"Î# , "Î#  , "Î# , "Î# œ +#Î#  , #Î# œ +  ,

82.

ˆB$Î#  C"Î# ‰ œ ˆB$Î#  C"Î# ‰ˆB$Î#  C"Î# ‰ œ B$Î# B$Î#  B$Î# C"Î#  B$Î# C"Î#  C"Î# C"Î# #

œ B'Î#  #B$Î# C"Î#  C#Î# œ B$  #B$Î# C"Î#  C

20

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SECTION 0.4 È$  "

œ

È&  #

œ

# ŠÈ $  " ‹

ŠÈ $ ‹  " #

œ

# ŠÈ $  " ‹

ŠÈ & ‹ 

œ

È&  #

œ

$BŠÈ(  #‹

# ŠÈ $  " ‹

œ È$  "

È$  "

œ

È$  " È$  "

È&  #

œ

È&  # È&  #

85.

È(  #

œ

86.

"%CŠÈ#  $‹ "%CŠÈ#  $‹ È#  $ "%C "%C œ † œ œ œ #CŠÈ#  $‹ # È#  $ È#  $ È#  $ #* # È Š #‹  $

83.

84.

#

"

$B

#

"

È(  #

È(  # È(  # $B

œ

#

" ŠÈ &  # ‹ #

##

$BŠÈ(  #‹ ŠÈ ( ‹  # # #

$"

&%

œ

œ

#

È&  #

(%

"

œ È&  #

œ

$BŠÈ(  #‹ $

87.

BŠB  È$‹ BŠB  È$‹ B B B  È$ œ † œ œ # B#  $ B  È$ B  È$ B  È$ B#  ŠÈ$‹

88.

#C  È(

89.

ŠC  È#‹ŠC  È#‹ C  È# C  È# C  È# C#  #CÈ#  # œ † œ œ # C#  # C  È# C  È# C  È# C #  ŠÈ # ‹

90.

91.

C

œ

œ BŠÈ(  #‹

CŠ#C  È(‹ CŠ#C  È(‹ #C  È( œ œ # %C#  ( #C  È( #C  È( #C#  ŠÈ(‹ C

ŠB  È$‹ŠB  È$‹ B  È$ B  È$ B  È$ B#  #BÈ$  $ œ † œ œ # B#  $ B  È$ B  È$ B  È$ B#  ŠÈ$‹

È #  È '  È $  ŠÈ $ ‹ È#  È$ "  È$ È#  È'  È$  $ œ † œ œ # "$ "  È$ "  È$ "  È$ " #  ŠÈ $ ‹

È#  È$

#

œ œ œ

21

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È#  È'  È$  $

# È Š #  È '  È $  $‹ # È$  $  È#È' #


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SECTION 0.4

92.

È $  È '  È #  ŠÈ # ‹ È$  È# "  È# È$  È'  È#  # œ † œ œ # "# "  È# "  È# "  È# " #  ŠÈ # ‹

È$  È#

#

œ

È$  È'  È#  #

" œ  ŠÈ $  È '  È #  # ‹

93.

ÈB  ÈC

94.

È#B  C

95.

È#  "

96.

ÈB  $

ÈB  ÈC È#B  C

#

$

œ È'  È#  È$  #

œ

œ

ÈB#  ÈBC  ÈBC  ÈC# ÈB  ÈC ÈB  ÈC B  #ÈBC  C † œ œ # # ÈB  ÈC ÈB  ÈC BC ˆÈB‰  ˆÈC‰

È%B#  CÈ#B  CÈ#B  C# È#B  C È#B  C #B  #CÈ#B  C# † œ œ # È#B  C È#B  C #B  C# ŠÈ#B‹  C#

ŠÈ # ‹  " # È#  " È#  " #" " œ † œ œ œ È#  " # # ŠÈ #  " ‹ # ŠÈ #  " ‹ # ŠÈ #  " ‹ #

œ

ÈB  $ ÈB  $ ˆÈB‰#  $# B* † œ œ È ˆ È ‰ ˆ È $ B$ $ B$ $ B  $‰

C #  ŠÈ $ ‹ C  È$ C  È$ C  È$ C#  $ œ † œ œ C  È$ C  È$ C  È$ C#  CÈ$  CÈ$  È* C#  #CÈ$  $ #

97.

ˆÈ+‰#  ŠÈ, ‹ È+  È, È+  È, È+  È, +, œ † œ œ È+#  È+,  È+,  È, # È+  È, È+  È, È+  È, +  #È+,  , #

98.

99.

ÈB  $  ÈB $

œ

# ÈB  $  ÈB ÈB  $  ÈB ˆÈB  $‰  ˆÈB‰# † œ ÈB  $  ÈB $ $ˆÈB  $  ÈB‰ B$B œ È $ˆ B  $  ÈB‰ $ " œ œ È È $ˆ B  $  ÈB‰ B  $  ÈB

22

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SECTION 0.4

100.

È#  2  È# 2

# ˆÈ#  2‰  ŠÈ#‹ È#  2  È# È#  2  È# œ † œ È#  2  È# 2 2 ŠÈ #  2  È # ‹

œ œ

#2# 2 ŠÈ #  2  È # ‹

$'+# , $ #+ œ #+#" , $' œ #+" , $ œ $ ' ")+, ,

102.

%&<# =& >$ & #' &# $) & % $ & &=$ œ  < = > œ  < = > œ  #(<' =# >) $ $ $<% >&

103.

"'B' C% D * # # #D * œ  B'* C%' D *! œ  B$ C# D * œ  $ # * ' ! #%B C D $ $ $B C

104.

$#7' 8% :# "' '' %( ## "' ! $ ! "' œ 7 8 : œ 7 8 : œ #'7' 8( :# "$ "$ "$8$ &B$ C#  "&B$ C% &B$ C# "&B$ C% œ  "!B# C$ "!B# C$ "!B# C$ B $BC œ  #C #

106.

2

2 ŠÈ #  2  È # ‹

101.

105.

#

œ

" È#  2  È#

*7% 8*  '7$ 8% *7% 8* '7$ 8% œ  "#7$ 8$ "#7$ 8$ "#7$ 8$ $78' 8 œ  % #

107.

#%B& C(  $'B# C&  "#BC #%B& C( $'B# C& "#BC #C$ $C " œ   œ  $ % $ & % & % & % & % '!B C '!B C '!B C '!B C & &B &B C

108.

*+$ , %  #(+# , %  ")+# , $ *+$ , % #(+# , % ")+# , $ + $ " œ   œ $ $ % # ( # ( # ( ")+ , ")+ , ")+ , ")+# , ( #, #, ,

109.

$B  # B  $ $B#  ""B  ' $B#  *B #B  ' #B  ' !

111.

B ( #B  & #B#  "*B  #B#  &B  "%B   "%B 

# #B&

110.

B $ $B  # $B#  ""B  ' $B#  #B *B  ' *B  ' !

112.

#B  & B  ( #B#  "*B  $& #B#  "%B  &B  $&  &B  $& !

$( $( $& #

23

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SECTION 0.4 113.

#B#  #B  #  B  " #B$  !B#  !B  #B$  #B# #B#  !B  #B#  #B #B  #B 

$ B"

114.

" " " # $

B#  B  $  #B  ( #B$  *B#  "$B  #B$  (B#  #B#  "$B   #B#  (B 'B  'B 

" #B(

#! #! #! #" "

116.

B  #  B" B# $ B#  $ B$  #B#  %B  & B$  $B  #B#  B  &  #B# '  B"

& B#  #  B B$ # $ & % $ B  # B  !B  #B  $B#  !B  * B&  #B# $  #B  B#  !B  *  #B$ %  B# &

118.

B#  #  B$$$ B  $ B&  !B%  #B$  $B#  !B  * B&  $B# $  #B  !B  *  #B$ ' $

119.

B%  #B$  %B#  )B  "' B  # B&  !B%  !B$  !B#  !B  $# B&  #B% #B%  !B$ #B%  %B$ %B$  !B# %B$  )B# )B#  !B )B#  "'B "'B  $# "'B  $# !

120.

121.

'B#  B "# 'B  ""B  "! $'B%  (#B$  "#"B#  "%#B  "#! $'B%  ''B$  '!B# 'B$  '"B#  "%#B 'B$  ""B#  "!B  (#B#  "$#B  "#!  (#B#  "$#B  "#! !

115.

B $ B  B  " B$  #B#  %B  $ B$  B#  B  $B#  $B  $  $B#  $B  $ !

117.

#

#

$

B$  B#  B  " B  " B%  !B$  !B#  !B  " B%  B$  B$  !B#  B$  B# B#  !B B#  B  B"  B" !

#

24

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SECTION 0.4 122.

'B#  ""B  "! 'B  B  "# $'B%  (#B$  "#"B#  "%#B  "#! $'B%  'B$  (#B# ''B$  %*B#  "%#B ''B$  ""B#  "$#B  '!B#  "!B  "#!  '!B#  "!B  "#! ! #

123. Area œ length † width œ B  &B  # ft# œ B#  #B  &B  "! ft# œ B#  $B  "! ft# 124.

" † base † height # " B#  $B  %! œ B  ) † height # #ˆB#  $B  %!‰ œ B  ) † height Area œ

#B#  'B  )! œ B  ) † height #B#  'B  )! œ height B)

125. Volume œ 6 † A † 2 œ "#  #B"#  #BB in.$ œ ˆ"%%  %)B  %B# ‰B in.$ œ ˆ"%%B  %)B#  %B$ ‰ in.$

#B  "! B  ) #B#  'B  )! #B#  "'B  "!B  )!  "!B  )! ! The height is #B  "! ft.

. $B#  "*B  #! œ < $B  % B & $B  % $B#  "*B  #! $B#  %B "&B  #! "&B  #! ! > œ B&

126. > œ

127. +  ,  -# œ +  ,  -+  ,  - œ ++  ,  -  , +  ,  -  -+  ,  - œ +#  +,  +-  +,  , #  ,-  +-  ,-  -# œ +#  , #  -#  #+,  #,-  #+128. +  ,  -  . # œ +  ,  -  . +  ,  -  .  œ ++  ,  -  .   , +  ,  -  .   -+  ,  -  .   . +  ,  -  .  œ +#  +,  +-  +.  +,  , #  ,-  ,.  +-  ,-  -#  -.  +.  ,.  -.  .# œ +#  , #  -#  . #  #+,  #+-  #+.  #,-  #,.  #-. 129. Answers may vary.

130. Multiply the numeratior and denominator by the conjugate of the numerator ˆÈB  #‰. 131. Check the formula with + œ " and , œ #.

132. Check the formula with + œ $ and , œ %.

133. *$Î# œ ˆ*"Î# ‰ œ $$ œ #( $

25

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SECTION 0.4 134. Œ

)  "#&

135. Œ

'#&B%  "'C)

#Î$

œŒ

$Î%

"#&  )

#Î$

œ Œ

'#&$Î% B% 

"#&  )

$Î%

œ

"'$Î% C) $Î%

œ

"Î$

#

& #&  œ Œ# œ %

"#&B$ )C'

#

136. È)!B% œ È"'B% È& œ %B# È&

$ $ $ $ $ $ $ $ $ $ 137. È"'+, %  , È &%+, œ È), $ È #+,  , È #(È #+, œ #, È #+,  , $È #+, œ #, È #+,  $, È #+, $ È œ , #+, % % % % % % % % % 138. BÈ ",#)!B  È)!B& œ BÈ #&'È &B  È"'B% È &B œ %BÈ &B  #BÈ &B œ 'BÈ &B

Exercises 0.5

(page 61)

1.

factor

2.

integer, prime

3.

+B  ,B œ B+  , 

4.

B#  C# œ B  CB  C

5. 7. 9. 11. 13. 15. 17. 18. 19. 21. 23. 24. 25.

B#  #BC  C# œ B  CB  C œ B  C#

B$  C$ œ B  CB#  BC  C# 

$B  ' œ $B  #

6. 8. 10.

)B#  %B$ œ %B# #  B

(B# C#  "%B$ C# œ (B# C# "  #B

+B  C  , B  C œ B  C+  , 

12. 14. 16.

B#  #BC  C# œ B  CB  C œ B  C#

B$  C$ œ B  CB#  BC  C# 

&C  "& œ &C  $

*C$  'C# œ $C# $C  #

#&C# D  "&CD# œ &CD&C  $D

, B  C  +B  C œ B  C,  +

%+  ,  "#+#  $+, œ %+  ,  $+%+  ,  œ "%+  ,   $+%+  ,  œ %+  , "  $+ B#  %B  BC  %C œ BB  %  CB  % œ B  %B  C %B#  * œ #B#  $# œ #B  $#B  $

%  *<# œ ##  $<# œ #  $<#  $<

20. 22.

$'D #  %* œ 'D #  (# œ 'D  ('D  (

"'  %*B# œ %#  (B# œ %  (B%  (B

)"B%  " œ *B#   "# œ *B#  "*B#  " œ *B#  "$B  "$B  " #

)"  B% œ *#  B#  œ *  B# *  B#  œ *  B# $  B$  B #

B  D #  #& œ B  D #  &# œ B  D  &B  D  &

26.

B  C#  * œ B  C#  $# œ B  C  $B  C  $

26

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SECTION 0.5 27. 29. 31.

B#  )B  "' œ B  %B  % œ B  %#

, #  "!,  #& œ ,  &,  & œ ,  &#

28. 30.

C#  "%C  %* œ C  (C  ( œ C  (#

7#  %78  %8# œ 7  #87  #8

32.

<#  )<=  "'=# œ <  %=<  %=

"#B#  BC  'C# œ %B  $C$B  #C

34.

)B#  "!BC  $C# œ %B  C#B  $C

œ 7  #8

#

33.

+#  "#+  $' œ +  '+  ' œ +  '#

œ <  %=#

35.

B#  "!B  #": + œ ", , œ "!, - œ #" key number œ +- œ "#" œ #" B#  "!B  #" œ B#  (B  $B  #" œ BB  (  $B  ( œ B  (B  $

36.

B#  (B  "!: + œ ", , œ (, - œ "! key number œ +- œ ""! œ "! B#  (B  "! œ B#  &B  #B  "! œ BB  &  #B  & œ B  &B  #

37.

B#  %B  "#: + œ ", , œ %, - œ "# key number œ +- œ ""# œ "# B#  %B  "# œ B#  'B  #B  "# œ BB  '  #B  ' œ B  'B  #

38.

B#  #B  '$: + œ ", , œ #, - œ '$ key number œ +- œ "'$ œ '$ B#  #B  '$ œ B#  *B  (B  '$ œ BB  *  (B  * œ B  *B  (

39.

':#  (:  $: + œ ', , œ (, - œ $ key number œ +- œ '$ œ ") ':#  (:  $ œ ':#  *:  #:  $ œ $:#:  $  #:  $ œ #:  $$:  "

40.

%; #  "*;  "#: + œ %, , œ "*, - œ "# key number œ +- œ %"# œ %) %; #  "*;  "# œ %; #  $;  "';  "# œ ; %;  $  %%;  $ œ %;  $;  %

41. 42. 43. 44. 45. 46. 47. 48. 49.

>$  $%$ œ >$  ($ œ >  (c>#  >(  (# d œ >  (>#  (>  %*

<$  )=$ œ <$  #=$ œ <  #=<#  <#=  #=# ‘ œ <  #=<#  #<=  %=#  )D $  #( œ #D $  $$ œ #D  $#D #  #D $  $# ‘ œ #D  $%D#  'D  *

"#&+$  '% œ &+$  %$ œ &+  %&+#  &+%  %# ‘ œ &+  %#&+#  #!+  "' $+# ,-  '+, # -  *+,-# œ $+,-+  #,  $-

&B$ C$ D $  #&B# C# D #  "#&BCD œ &BCDB# C# D#  &BCD  #&

$B$  $B#  B  " œ $B# B  "  "B  " œ B  "$B#  "

%B  'BC  *C  ' œ #B#  $C  $$C  # œ $C  ##B  $

#>BC  #->B  $>C  $-> œ >#BC  #-B  $C  $- œ >c#BC  -  $C  -d œ >C  -#B  $ 27

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SECTION 0.5 50. 51. 52.

53.

55.

56.

57. 58.

#+B  %+C  ,B  #,C œ #+B  #C  , B  #C œ B  #C#+  , 

+B  ,B  +C  ,C  +D  ,D œ B+  ,   C+  ,   D+  ,  œ +  , B  C  D

'B# C$  ")BC  $B# C#  *B œ $Bˆ#BC$  'C  BC#  $‰ œ $B#CˆBC#  $‰  "ˆBC#  $‰‘ œ $BˆBC#  $‰#C  "

B#  C  D # œ cB  C  D dcB  C  D d œ B  C  D B  C  D 

54.

D#  C  $# œ cD  C  $dcD  C  $d œ D  C  $D  C  $

B  C#  B  C# œ cB  C  B  CdcB  C  B  Cd œ B  C  B  CB  C  B  C œ #B#C œ %BC #+  $#  #+  $# œ c#+  $  #+  $dc#+  $  #+  $d œ #+  $  #+  $#+  $  #+  $ œ %+' œ #%+

B%  C% œ B#   C#  œ B#  C# B#  C#  œ B#  C# B  CB  C #

#

D %  )" œ D #   *# œ D#  *D#  * œ D#  *D#  $#  œ D#  *D  $D  $ #

$B$ C  $BC œ $BCˆB#  "‰ œ $BCB  "B  "

59.

$B#  "# œ $B#  % œ $B  #B  #

60.

61.

")BC#  )B œ #Bˆ*C#  %‰ œ #B$C  #$C  #

62.

#(B#  "# œ $ˆ*B#  %‰ œ $$B  #$B  #

63.

B#  #B  "& Ê prime

64.

B#  B  # Ê prime

65.

"&  #+  #%+# œ #%+#  #+  "& œ '+  &%+  $

66.

$#  ')B  *B# œ *B#  ')B  $# œ *B  %B  )

67. 69. 70. 71. 72. 73. 74.

'B#  #*BC  $&C# œ $B  (C#B  &C

68.

"!B#  "(BC  'C# œ &B  'C#B  C

"#:#  &):;  (!; # œ #':#  #*:;  $&; #  œ #':  $&; :  ;  $B#  'BC  *C# œ $B#  #BC  $C#  œ $B  $CB  C

'7#  %(78  $&8# œ '7#  %(78  $&8#  œ '7  &87  (8 "%<#  ""<=  "&=# œ "%<#  ""<=  "&=#  œ (<  &=#<  $=

'B$  #$B#  $&B œ B'B#  #$B  $& œ B'B  (B  & C$  C#  *!C œ CC#  C  *! œ CC  "!C  *

28

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SECTION 0.5 75. 76. 77. 79. 81. 83.

85.

'B%  ""B$  $&B# œ B# 'B#  ""B  $& œ B# #B  ($B  &

"#B  "(B#  (B$ œ (B$  "(B#  "#B œ B(B#  "(B  "# œ BB  $(B  %

B%  #B#  "& œ B#  &B#  $

B%  B#  ' œ B#  $B#  #

78.

+#8  #+8  $ œ +8  $+8  "

+#8  '+8  ) œ +8  %+8  #

80.

'B#8  (B8  # œ $B8  ##B8  "

*B#8  *B8  # œ $B8  #$B8  "

82.

%B#8  *C#8 œ #B8 #  $C8 # œ #B8  $C8 #B8  $C8 

"!C#8  ""C8  ' œ &C8  ##C8  $

84.

)B#8  #B8  $ œ %B8  $#B8  "

86.

"'C%8  #&C#8 œ C#8 ˆ"'C#8  #&‰ œ C#8 %C8 #  &# ‘

œ C#8 %C8  &%C8  &

87. 88. 89.

90.

91.

#B$  #,!!! œ #B$  ",!!! œ #B$  "!$  œ #B  "!B#  "!B  "!!

$C$  '%) œ $C$  #"' œ $C$  '$  œ $C  'C#  'C  $'

B  C$  '% œ B  C$  %$ œ cB  C  %dB  C#  %B  C  %# ‘ œ B  C  %ˆB#  #BC  C#  %B  %C  "'‰

B  C$  #( œ B  C$  $$ œ cB  C  $dB  C#  $B  C  $# ‘ œ B  C  $ˆB#  #BC  C#  $B  $C  *‰

# # '%+'  C' œ ˆ)+$ ‰  ˆC$ ‰ œ ˆ)+$  C$ ‰ˆ)+$  C$ ‰ œ #+  Cˆ%+#  #+C  C# ‰#+  Cˆ%+#  #+C  C# ‰

œ #+  C#+  Cˆ%+#  #+C  C# ‰ˆ%+#  #+C  C# ‰

92. 93. 94.

+'  , ' œ +#   , #  œ +#  , # Š+#   +# , #  , #  ‹=+#  , # +%  +# , #  , %  $

$

#

#

+$  , $  +  , œ +  , +#  +,  , #   +  , " œ +  , +#  +,  , #  " +#  C#   &+  C œ +  C+  C  &+  C œ +  C+  C  &

95.

'%B'  C' œ ˆ%B# ‰  ˆC# ‰ œ ˆ%B#  C# ‰Šˆ%B# ‰  %B# C#  ˆC# ‰ ‹

96.

D #  'D  *  ##&C# œ D  $D  $  ##&C# œ D  $#  "&C# œ D  $  "&CD  $  "&C

$

$

#

œ ˆ%B#  C# ‰ˆ"'B%  %B# C#  C% ‰

29

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SECTION 0.5 97.

B#  'B  *  "%%C# œ B  $B  $  "%%C# œ B  $#  "#C# œ B  $  "#CB  $  "#C

98.

B#  #B  *C#  " œ B#  #B  "  *C# œ B  "B  "  *C#

99.

+  , #  $+  ,   "! œ c+  ,   &dc+  ,   #d œ +  ,  &+  ,  #

œ B  "#  $C# œ B  "  $CB  "  $C

100. #+  , #  &+  ,   $ œ c#+  ,   "dc+  ,   $d œ #+  #,  "+  ,  $ 101. B'  (B$  ) œ B$  )B$  " œ B  #B#  #B  %B  "B#  B  "

102. B'  "$B%  $'B# œ B# B%  "$B#  $' œ B# B#  *B#  % œ B# B  $B  $B  #B  # 103. B%  B#  " œ B%  #B#  "  B# œ ˆB#  "‰ˆB#  "‰  B#

104. B%  $B#  % œ B%  %B#  %  B# œ ˆB#  #‰ˆB#  #‰  B#

105. B%  (B#  "' œ B%  )B#  "'  B# œ ˆB#  %‰ˆB#  %‰  B#

106. C%  #C#  * œ C%  'C#  *  %C# œ ˆC#  $‰ˆC#  $‰  %C#

# œ ˆB#  "‰  B# œ ˆB#  "  B‰ˆB#  "  B‰

# œ ˆB#  %‰  B# œ ˆB#  %  B‰ˆB#  %  B‰

# œ ˆB#  #‰  B# œ ˆB#  #  B‰ˆB#  #  B‰

# œ ˆC#  $‰  #C# œ ˆC#  $  #C‰ˆC#  $  #C‰

107. %+%  "  $+# œ %+%  %+#  "  +# œ ˆ#+#  "‰ˆ#+#  "‰  +# œ ˆ#+#  "‰  +# œ ˆ#+#  "  +‰ˆ#+#  "  +‰ #

108. B%  #&  'B# œ B%  "!B#  #&  %B# œ ˆB#  &‰ˆB#  &‰  %B#

œ ˆB#  &‰  #B# œ ˆB#  &  #B‰ˆB#  &  #B‰ #

% $ % $ 1<  1< $ " $ # % œ 1ˆ<"$  <#$ ‰ $ % œ 1<"  <# ˆ<"#  <" <#  <## ‰ $

109. Z œ

111-114. Answers may vary.

115. $B  # œ #ˆ $B  ## ‰ œ #ˆ $# B  "‰ # 117. B#  #B  % œ #Š B#  #

#B #

 %# ‹

œ #ˆ "# B#  B  #‰

110. 0 œ "%%  "'># œ "'ˆ*  ># ‰ œ "'$  >$  >

116. &B  $ œ &ˆ &B  $& ‰ œ &ˆB  $& ‰ & 118. $B#  #B  & œ $Š $B$  #

#B $

 &$ ‹

œ $ˆB#  #$ B  &$ ‰

30

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SECTION 0.5 119. +  , œ +ˆ ++  +, ‰ œ +ˆ"  +, ‰

121. B  B"Î# œ B"Î# ˆB""Î#  B"Î#"Î# ‰ œ B"Î# ˆB"Î#  "‰ 123. #B  È#C œ È#

È#C #B  È# È# 

œ È#ŠÈ#B  C‹

125. +, $Î#  +$Î# , œ +, 

+, $Î# +$Î# ,  +, +, 

œ +, ˆ, "Î#  +"Î# ‰

120. +  , œ , ˆ +,  ,, ‰ œ , ˆ +,  "‰

122. B$Î#  B"Î# œ B"Î# ˆB$Î#"Î#  B"Î#"Î# ‰ œ B"Î# B  "

124. È$+  $, œ È$

È$+ È$



$, È$ 

œ È$Š+  È$, ‹

126. +, #  , œ , " Œ

+, # ,  "  , " , œ , " ˆ+, $  , # ‰

127. B#  B  '  BC  #C œ B  $B  #  CB  # œ B  #B  $  C

128. #B#  &B  #  BC  #C œ #B  "B  #  CB  # œ B  ##B  "  C

129. +%  #+$  +#  +  " œ +# ˆ+#  #+  "‰  +  " œ +# +  "+  "  "+  " œ +  "+# +  "  "‘ œ +  "ˆ+$  +#  "‰

130. +%  +$  #+#  +  " œ +# ˆ+#  +  #‰  +  " œ +# +  #+  "  "+  " œ +  "+# +  #  "‘ œ +  "ˆ+$  #+#  "‰

131. The number 1 is neither prime nor composite.

132. c#, $ Ê

oq񕕄qp #

$

+$  +#  +* +) +"( # $ œ $ œ "& œ + # $ & + + +  + 

133. B$ B#  œ B&  œ B#!

134.

135. Œ

136. È#!B& œ È%B% È&B œ #B# È&B

%

%

$B% B$  œ" 'B# B% !

$

%

137. È#!B  È"#&B œ È%È&B  È#&È&B œ #È&B  &È&B œ $È&B 138.

$ $ $ $ $ È * $È * $È * È œ † œ œ œ $ * $ $ $ $ È$ È$ È* È#( $

31

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SECTION 0.6

Exercises 0.6

(page 71)

1.

numerator

2.

denominator

3.

+. œ ,-

4.

zero

5.

+,.

6.

+. ,-

7.

+,

8.

+,

9.

)B ? "'B œ $C 'C ? )B † 'C œ $C † "'B

10.

$B# ? "#C# œ %C# "'B# ? %C# † "#C# $B# † "'B# œ %)B% Á %)C% NOT EQUAL

%)BC œ %)BC EQUAL 11.

#&BCD ? &!+# ,œ "#+, # #%BCD ? #&BCD † #%BCD œ "#+, # - † &!+# ,-

12.

'!!B# C# D # Á '!!+$ , $ -# NOT EQUAL

"&<=# ? $(Þ&+$ œ %<=# "!+$ ? "&<=# † "!+$ œ %<=# † $(Þ&+$ "&!<=# +$ œ "&!<=# +$ EQUAL

13.

(+# , + † (+, + (+, + œ œ † œ #"+, # $, † (+, $, (+, $,

14.

$&:$ ; # &; † (:$ ; &; (:$ ; &; œ œ † œ % $ %*: ; (: † (: ; (: (:$ ; (:

15.

%B # %B † # )B † œ œ ( &+ ( † &+ $&+

16.

&C % &C † % #!C "! † œ œ œ #D C# #D † C# #C# D CD

17.

)7 $7 )7 "!8 )!78 "' ƒ œ † œ œ &8 "!8 &8 $7 "&78 $

18.

"&: &: "&: "'; # #%!:; # ƒ œ † œ œ '; # ); "'; ); &: %!:;

19.

$D #D $D  #D &D D  œ œ œ &&&&-

20.

(+ $+ (+  $+ %+ +  œ œ œ %, %, %, %, ,

21.

"&B# C B# C "%B# C #B# C  œ œ (+# , $ (+# , $ (+# , $ +# , $

22.

)<=># (<=># "&<=># <=  œ œ % "&7% ># "&7% ># "&7% ># 7

23.

#B  % #B  # # œ œ B#  % B  #B  # B#

24.

B#

B#  "' B  %B  % B% œ œ  )B  "' B  %B  % B%

25.

%  B# #  B#  B B# œ œ B#  &B  ' B  $B  # B$

27.

'B$  B#  "#B B'B#  B  "# B#B  $$B  % $B  % œ œ œ $ # # %B  %B  $B B%B  %B  $ B#B  $#B  " #B  "

26.

#&  B# &  B&  B œ B#  "!B  #& B  &B  & &B B& œ œ B& B&

32

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SECTION 0.6 28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

'B%  &B$  'B# B# 'B#  &B  ' B# #B  $$B  # B#B  $$B  # œ œ œ $ # # #B  (B  "&B B#B  (B  "& B#B  $B  & #B  $B  & B#

B$  ) B$  #$ B#  #B  % B  #B#  #B  % œ œ œ  +B  #B  #+ BB  +  #B  + B  +B  # B+

BC  #B  $C  ' BC  #  $C  # C  #B  $ C# œ œ œ # B  $B#  $B  * B$  #( B$  $$ B  $B  * B  "B  " B#  " B# B# BB  " † # œ † œ B  "B  " B B  #B  " B B"

C#  #C  " C# C  "C  " C# C" † # œ † œ C  #C  " C C C# C C $B#  (B  # B#  B B" $B  "B  # BB  " † # œ † œ B#  #B $B  B BB  # B$B  " B

B#  B #B#  B  $ BB  " #B  $B  " † œ † œ" #B#  $B B#  " B#B  $ B  "B  " B#  B B#  " BB  " B  "B  " BB  "# † œ † œ B" B# B" B# B#

B#  &B  ' B  # B# B# B  #B  $ † # œ † œ # B  'B  * B  % B  $B  $ B  #B  # B  $B  # #B#  $# B#  "' #B#  $# # #B#  "' # " ƒ œ † # œ † # œ ) # ) B  "' ) B  "' #

B#  B  ' B#  % B#  B  ' B#  * B  $B  # B  $B  $ ƒ œ † # œ † # # # B  'B  * B  * B  'B  * B  % B  $B  $ B  #B  # B  $# œ B  $B  #

D #  D  #! D #  #& D#  D  #! D  & D  &D  % D& ƒ œ † # œ † # D % D& D#  % D  #& D  #D  # D  &D  & D% œ D  #D  #

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SECTION 0.6 40.

41.

42.

43.

44.

45.

+B  ,B  +  , B#  " +B  ,B  +  , B#  #B  " ƒ # œ # † # # +  #+,  , B  #B  " +  #+,  , # B#  " B+  ,   "+  ,  B  "B  " œ † +  , +  ,  B  "B  " +  , B  " B  "B  " B" † œ œ +  , +  ,  B  "B  " +, $B#  &B  # 'B#  "$B  & $B#  &B  # #B$  &B# ƒ œ † # $ # $ # $ # B  #B #B  &B B  #B 'B  "$B  & $B  "B  # B# #B  & œ † œ" $B  "#B  & B# B  # B#  "$B  "# #B#  B  $ B#  "$B  "# )B#  "%B  & ƒ œ † )B#  'B  & )B#  "%B  & )B#  'B  & #B#  B  $ B  "#B  " %B  &#B  " B  "##B  " œ † œ %B  &#B  " #B  $B  " #B  "#B  $ B#  (B  "# B#  $B  "! B$  %B#  $B † † # B$  B#  'B B#  #B  $ B  B  #! B  $B  % B  &B  # BB  $B  " œ † † œ" BB  $B  # B  $B  " B  &B  %

BB  #  $ BB  "  # B#  #B  $ B#  B  # † œ # † # BB  (  $B  " BB  (  $B  " B  (B  $B  $ B  (B  $B  $ B#  #B  $ B#  B  # œ # † B  %B  $ B#  %B  $ B# B  $B  " B  #B  " œ † œ B$ B  $B  " B  $B  " B#  #B  $ $B  ) B#  'B  & B#  #B  $ $B  ) (B#  $$B  "! † ƒ œ † † #"B#  &!B  "' B  $ (B#  $$B  "! #"B#  &!B  "' B  $ B#  'B  & B  $B  " $B  ) (B  #B  & œ † † (B  #$B  ) B  $ B  &B  " B& œ B&

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SECTION 0.6 46.

47.

49. 51.

53.

54.

55.

B$  #( B#  %B  $ B#  B  ' B$  #( B#  %B  $ B#  $B  * ƒŒ ƒ # ƒŒ † # œ #  # # B % B  #B B  $B  * B % B#  #B B B' $ # B  #( B  $B  " B  $B  * œ # ƒŒ †  B  $B  # B % BB  # B$  #( B  "B#  $B  * œ # ƒ B % BB  #B  # # B  $B  $B  * BB  #B  # œ † B  #B  # B  "B#  $B  * BB  $ œ B" $ B# $B# B&  œ œ B$ B$ B$ B$

%B % %B  % %B  "  œ œ œ% B" B" B" B"

# " # " "  œ  œ &B B& B& B& B&

48.

50. 52.

$ B# $B# B&  œ œ B" B" B" B"

'B $ 'B  $ $#B  "  œ œ B# B# B# B#

$ # $ # &  œ  œ B' 'B B' B' B'

$ # $B  " #B  " $B  $ #B  #  œ  œ  B" B" B  "B  " B  "B  " B  "B  " B  "B  " &B  " œ B  "B  " $ B $B  % BB  % $B  "# B#  %B  œ  œ  B% B% B  %B  % B  %B  % B  %B  % B  %B  % B#  (B  "# œ B  %B  %

+$ + +$ +  œ  +#  (+  "# +#  "' +  $+  % +  %+  % " + œ  +  % +  %+  % "+  % + œ  +  %+  % +  %+  % +% + œ  +  %+  % +  %+  % #+  % #+  # œ œ +  %+  % +  %+  %

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SECTION 0.6 56.

57.

58.

59.

60.

+ # + #  œ  +  #+  " +  %+  " +#  +  # +#  &+  % ++  % #+  # œ  +  #+  "+  % +  %+  "+  # +#  %+ #+  % œ  +  #+  "+  % +  #+  "+  % +#  #+  % œ +  #+  "+  % B#

,#

B " B " B "B  #  œ  œ  % B# B  #B  # B  # B  #B  # B  #B  # B B# œ  B  #B  # B  #B  # # œ B  #B  #

,# % ,# % , # ,  %,  #  # œ  œ  ,  #,  # , ,  #  % ,  #, , ,  #,  # , ,  #,  # ,$ %,  ) œ  , ,  #,  # , ,  #,  # , $  %,  ) œ , ,  #,  #

$B  # B $B  # B  œ  B#  #B  " B#  " B  "B  " B  "B  " $B  #B  " BB  " œ  B  "B  "B  " B  "B  "B  " $B#  &B  # B#  B œ  B  "B  "B  " B  "B  "B  " #B#  'B  # #B#  $B  " œ œ B  "B  "B  " B  "# B  "

#> >" #> >" #>> >  ">  &  œ  œ  >#  #& >#  &> >  &>  & >>  & >>  &>  & >>  &>  & #># >#  %>  & œ  >>  &>  & >>  &>  & >#  %>  & œ >>  &>  &

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SECTION 0.6 61.

62.

63.

64.

65.

# " # $ œ C  "C  " C#  " C" # œ C  "C  " # œ C  "C  " $C#  C  # œ C  "C  " #

>#

$ "  " C" $C  "C  " "C  "   "C  "C  " C  "C  " $C#  $ C"   C  "C  " C  "C  " $C  #C  " $C  # œ œ C  "C  " C" 

% " # % "  œ   % ># " >  #>  # >  # #>  #>  # % ">  # œ   ">  #>  # >  #>  # >  #>  # #>#  ) % ># œ   >  #>  # >  #>  # >  #>  # #>#  >  ' #>  $>  # #>  $ œ œ œ >  #>  # >  #>  # >#

" $ $B  # " $ $B  #   # œ   B# B# B % B  # B  # B  #B  # "B  # $B  # $B  # œ   B  #B  # B  #B  # B  #B  # B# $B  ' $B  # œ   B  #B  # B  #B  # B  #B  # B# " œ œ B  #B  # B#

B & $$B  " B & *B  $   œ   B$ B$ B#  * B  $ B  $ B  $B  $ BB  $ &B  $ *B  $ œ   B  $B  $ B  $B  $ B  $B  $ B#  $B &B  "& *B  $ œ   B  $B  $ B  $B  $ B  $B  $ B#  (B  "# B  $B  % B% œ œ œ B  $B  $ B  $B  $ B$ Œ

" " B$ "B  $ "B  #  œŒ  † † B# B$ #B B  #B  $ B  $B  # B$ B# œŒ  † B  #B  $ B  #B  $ #B  & B$ #B  & œ † œ B  #B  $ #B #BB  #

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B$ #B B$ #B


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SECTION 0.6 66.

67.

68.

69.

Œ

" " " "B  # "B  " B#  œŒ  ƒ † B" B# B# B  "B  # B  #B  " " B# B" B# œŒ  † B  "B  # B  "B  # " $ B# $ œ † œ B  "B  # " B"

$B B $B  " $B B $B  "   œ   # B  % B  % "'  B B  % B  % %  B%  B $B B $B  " œ   B  % B  % B  %B  % $BB  % BB  % $B  " œ   B  %B  % B  %B  % B  %B  % B#  %B $B  " $B#  "#B œ   B  %B  % B  %B  % B  %B  % #B#  "*B  " œ B  %B  % (B $B $B  " (B $B $B  "   œ   B  & &  B B#  #& B  & B  & B  &B  & %B $B  " œ  B  & B  &B  & %BB  & $B  " œ  B  &B  & B  &B  & %B#  #!B $B  " %B#  #$B  " œ  œ B  &B  & B  &B  & B  &B  & " # "   B#  $B  # B#  %B  $ B#  &B  ' " # " œ   B  #B  " B  $B  " B  #B  $ "B  $ #B  # "B  " œ   B  #B  "B  $ B  $B  "B  # B  #B  $B  " B$ #B  % B" œ   B  #B  "B  $ B  #B  "B  $ B  #B  "B  $ B  $  #B  %  B  " ! œ œ œ! B  #B  "B  $ B  #B  "B  $

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SECTION 0.6 70.

71.

72.

73.

75.

# # #D  #C # # #D  #C   œ   B  C B  D C  BD  B C  B D  B C  BD  B #D  B #C  B #D  #C œ   C  BD  B D  BC  B C  BD  B #D  #B #C  #B #D  #C œ   C  BD  B C  BD  B C  BD  B #D  #B  #C  #B  #D  #C ! œ œ œ! C  BD  B C  BD  B B#

$B  # %B#  # $B#  #&  #  #  B  #! B  #& B  "' $B  # %B#  # $B#  #& œ   B  &B  % B  &B  & B  %B  % $B  #B  &B  % %B#  #B  %B  % œ  â B  &B  %B  &B  % B  &B  &B  %B  % $B#  #&B  &B  &  B  %B  %B  &B  & $ # % # $B  &B  &)B  %! %B  '#B  $# œ  â B  &B  %B  &B  % B  &B  %B  &B  % $B%  "!!B#  '#&  B  &B  %B  &B  % $B$  &B#  &)B  %!  %B%  '#B#  $#  $B%  "!!B#  '#& œ B  &B  %B  &B  % B%  $B$  %$B#  &)B  '*( œ B  &B  %B  &B  %

)B#

$+ , '+,#

$B  # B% "  #   "!B  $ 'B  ""B  $ %B  " $B  # B% " œ   %B  "#B  $ $B  "#B  $ %B  " $B  #$B  " B  %%B  " "#B  $$B  " œ   %B  "#B  $$B  " $B  "#B  $%B  " %B  "#B  $$B  " *B#  $B  # %B#  "(B  % 'B#  ""B  $ œ   %B  "#B  $$B  " %B  "#B  $$B  " %B  "#B  $$B  " *B#  $B  #  %B#  "(B  %  'B#  ""B  $ (B#  $"B  " œ œ %B  "#B  $$B  " %B  "#B  $$B  " œ

$+# , +, #(

$+ '+$+ , # , ƒ # œ † œ , , , '+#-

œ

$+# , +, $+# , #( ƒ œ † œ )"+ " #( " +,

74.

76.

$># *B > ")B $?# @ %>

$?@

œ

$># > $># ")B ƒ œ † œ '> *B ")B *B >

œ

$?# @ $?@ $?# @ " ? ƒ œ † œ %> " %> $?@ %>

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SECTION 0.6 77.

BC +, CB +,

78.

B# &B' #B# C B# * #B# C

BC CB BC +, ƒ œ † œ " +, +, +, CB

œ

œ

B#  &B  ' B#  * B#  &B  ' #B# C ƒ œ † # #B# C #B# C #B# C B * œ

79.

80.

81.

" B



" C

BC

"" B

" B " B

BCŠ B"  C" ‹ BCBC

œ

œ

BCˆ B" ‰  BCŠ C" ‹ B# C#

83.

84.

 

85.

86.

88.

89.

CB B# C#

" C " C

œ

BCŠ B"  C" ‹ BCŠ B"  C" ‹

BCˆ B" ‰  BCŠ C" ‹ BCˆ B" ‰  BCŠ C" ‹

œ

82.

CB CB

" B " B

 

" C " C

œ

BCŠ B"  C" ‹ BCŠ B"  C" ‹

BCˆ B" ‰  BCŠ C" ‹

BCˆ B" ‰  BCŠ C" ‹ CB œ CB œ

%+ %+ # ‰ +,BŠ $+ +,Bˆ $+  %+B $+# B  %+$ , +# $B  %+,  ,  B ‹ ,  +,BŠ B ‹ œ œ œ œ " " " " " +B  , +B  ,  +B +,Bˆ ,  +B ‰ +,Bˆ , ‰  +,Bˆ +B ‰ #

" B# C#

œ

BC BCBC B# C# B# C# B# C# œ œ œ "" œ "" "" ""C  ""B ""C  B  C ‰ BCŠ "" BCˆ "" B  C ‹ B  BC Š C ‹

œ $+ , " ,

B  $B  # #B# C B# † œ # #B C B  $B  $ B$

B C

"

œ

B" B&

' B ' B

C# Š"  BC ‹

C# Š BC#  "‹ #

œ

œ

#

C# "  C# Š BC ‹

C# Š BC# ‹  C# " #

œ

C#  BC CC  B C œ œ B#  C# BC B  CB  C

BˆB  "  B' ‰ BB  B"  Bˆ B' ‰ B#  B  ' B# B  $B  # œ œ œ œ '‰ '‰ # ˆ ˆ B  &B  ' B# B  #B  $ B B& B BB  B&  B B

#D D #D  #D # $ œ ˆ $‰ œ " D D " D D "  D ˆ $D ‰ #D # œ D$

87.

$BC BC$BC $B# C# œ " œ " " "  BC BCŠ"  BC BC"  BCŠ BC ‹ ‹ œ

B  $  B" BˆB  $  B" ‰ B#  $B  " B#  $B  " œ œ œ œ " " " # "  B  $B B#  $B  " B  B  $ Bˆ B  B  $‰

$B B$B $B# " œ ˆ "‰ œ # B " B B B B B

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$B# C# BC  "


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SECTION 0.6 90.

91.

#B#  % &#B#  % "!B#  #! #&B#  "! &B#  "! œ œ œ %B œ %B "!  %B #&  #B &  #B # & &ˆ#  & ‰ B B# $ B#

 

# B" B B"

œ

B B  #B  "ˆ B#  $ B  #B  "ˆ B# 

# ‰ B" B ‰ B"

œ

B ‰ # ‰ B  #B  "ˆ B#  B  #B  "ˆ B" B ‰ $ ‰ B  #B  "ˆ B#  B  #B  "ˆ B"

B  "B  B  ## B  "$  B  #B B#  B  #B  % B#  $B  % œ œ # # $B  $  B  #B B  &B  $ œ

92.

#B B$ $ B$

 

" B# B B#

œ

#B B  $B  #ˆ B$  $ B  $B  #ˆ B$ 

" " œ " "B "

95.

$B  #"  #B  "" œ B  #"

" B

#B ‰ " ‰ B  $B  #ˆ B$  B  $B  #ˆ B# $ ‰ B ‰ B  $B  #ˆ B$  B  $B  #ˆ B# B  ##B  B  $" œ B  #$  B  $B #B#  %B  B  $ œ $B  '  B#  $B #B#  $B  $ #B#  $B  $ œ œ  B#  'B  ' B#  'B  '

œ

B" B œ " B" Bˆ"  B ‰

93.

œ

" ‰ B# B ‰ B#

$ B#



# B"

94.

œ

" B#

C" œ B"  C"

" C " B



" C

$ # ‰ B  #B  "ˆ B#  B" " B  #B  "ˆ B# ‰

œ

BCŠ B"  C" ‹

$ ‰ # ‰ B  #B  "ˆ B#  B  #B  "ˆ B" B" B  "$  B  ## œ B" $B  $  #B  % &B  " œ œ B" B"

œ

96.

#BB  $"  $B  #" œ B  $" B  #"

#B B$



$ B#

" B$B#

œ

#B B  $B  #ˆ B$ 

" 5"

" 

" 5#

œ

5" 5# "

5" 5# Š 5""



" 5# ‹

œ

B  ##B  B  $$ " œ #B#  %B  $B  * œ #B#  B  *

5" 5#

5" 5# Š 5"" ‹



$ ‰ B#

B  $B  #Š B$" B# ‹

œ

97.

BCŠ "C ‹

5" 5# Š 5"# ‹

œ

5" 5# 5#  5"

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œ

B CB


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SECTION 0.6 98.

99.

" " V"



" V#



" V$

œ

V" V# V$ "

V" V# V$ Š V"" 

102.

" V$ ‹

B

B $

V" V# V$

V" V# V$ Š V"" ‹  V" V# V$ Š V"# ‹  V" V# V$ Š V"$ ‹

œ

V" V# V$ V# V$  V" V$  V" V#

œ

$B $B B‰ œ ˆ $B $ " $

+, +, +, +, #+, #+, œ œ œ $ $+ œ ˆ $+ ‰ œ $   + $ %  $+ #  #+" # # # # # # # #  +ˆ # ‰ +

101.



B B B B œ œ œ " "   B " " "  $B" " $ "  Bˆ $ ‰ B

100.

" V#

œ

+

B  ""

œ œ B œ B ‰ œ B  "  B œ #B  " B" "  B" B  "ˆ"  B" "  "" " "  Bˆ" " ‰ B B "

"

C

C

œ #  ## # # C

"

C #

œ

C Š# #C ‹

C #

œ

#C #C#

B"

#C  #C

#C  #Š# 

#C #C# ‹

œ

B"

#C#  #C #C  ##  #C

#CC  " %C  %  #C #CC  " CC  " œ œ #$C  # $C  #

œ

107. Since ' ! !, k'k œ ' œ '.

103-106. Answers may vary.

108. If B ! !, then &  B € !. Then k&  Bk œ &  B. 109. Œ

B$ C#  B" C

$

110. #(B' 

#Î$

œŒ

B" C C# C C$ C* œ œ œ  Œ  Œ  B$ C# B$ B B% B"# $

œ #(#Î$ B' 

#Î$

$

$

œ ˆ#("Î$ ‰ B"#Î$ œ $# B% œ *B% #

111. È#!  È%& œ È%È&  È*È& œ #È&  $È& œ È&

112. #B#  %  $#B#  & œ #B#   #%  $#B#   $& œ #B#  )  'B#  "& œ %B#  (

Chapter 0 Review

(page 75)

1.

natural: $, ', )

2.

whole: !, $, ', )

3.

integers: ', $, !, $, ', )

4.

rational: ', $, !, "# , $, ', )

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CHAPTER 0 REVIEW 5.

irrational: 1, È&

6.

real: ', $, !, "# , $, 1, È&, ', )

7.

prime: $

8.

composite: ', )

9.

even integers: ', !, ', )

10.

odd integers: $, $

11.

associative property of addition

12.

commutative property of addition

13.

associative property of multiplication

14.

distributive property

15.

commutative property of multiplication

16.

commutative property of addition

17.

double negative rule

18.

oñqñqqñqñp "" "$ "( "*

19.

oñqñqñqñqñp ' ) "! "# "%

20.

$ ! B Ÿ &

21.

B € ! or B ! "

oqÐïïÓqp $ 22.

íïÑqqÒïî

&

"

!

Ð#, %Ó

oqÐïïÓqp #

23.

%

∞, # ∩ &, ∞ ∞, # &, ∞

∞, # ∩ &, ∞ 24.

∞, % ∪ Ò', ∞Ñ

íïïïÑqp # oqÐïïïî & oqÐïïÑqp &

#

íïÑqqÒïî %

25. 27.

29.

'

Since ' € !, k'k œ '.

Since "  È# ! !, ¹"  È #¹ œ Š"  È #‹ œ È #  ".

26. 28.

distance œ k(  &k œ k"#k œ "#

Since #& ! !, k#&k œ #& œ #&. Since È$  " € !, ¹ È $  "¹ œ È $  " .

43

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CHAPTER 0 REVIEW &+# œ &+&+

30.

&+$ œ &+++

31.

32.

$>>> œ $>$

33.

34.

8# 8% œ 8#% œ 8'

35.

:$  œ :$†# œ :'

36.

B$ C#  œ B$  C#  œ B"# C)

37.

Œ

38.

ˆ7$ 8! ‰# œ ˆ7$ † "‰# œ 7' œ

39.

:# ; # ;# ; #  ;' Œ  œ Œ # œ $ œ # # #: ):' #: 

40.

+& " œ +&) œ +$ œ $ ) + +

41.

+# Œ $ ,

42.

Œ

$B# C#  B# C#

43.

Œ

+$ , #  +, $

44.

$B$ C Œ  BC$

œŒ

Œ

$

45.

%

%

œŒ

#

#

%

œŒ

#

#7# 8!  %7# 8"

#, $,  œ #$,, œ ', # #

+% +%  +"#  œ # $ œ ' # , , ,  $

$

$

" 7'

#

$

œŒ

$

,$ ,'  œ % # + + #

B# C# B# C# C# C% C) œ œ œ  Œ  Œ  $B# C# $B# $ * #

#

#

+, $ ++$ +% +) œ œ œ  Œ  Œ  +$ , # ,#,$ ,& , "! #

#

#

BC$ C# C%  œŒ  œ % $ # $B C $B *B #

œ Œ

#

%7# 8" #7# 7# #7% )7"# œ  œ  œ   Œ  Œ  #7# 8! 8" 8! 8 8$ $

$

$

46.

B#  BC# œ $#  $$# œ *  $* œ *  #( œ *  #( œ ")

47.

',(&! œ 'Þ(&! ‚ "!$

48.

!Þ!!!#$ œ #Þ$ ‚ "!%

49.

%Þ) ‚ "!# œ %)!

50.

!Þ#& ‚ "!$ œ !Þ!!!#&

51.

%&,!!!$&!,!!! %Þ& ‚ "!% $Þ& ‚ "!&  %Þ& ‚ $Þ& ‚ "!% ‚ "!& "&Þ(& ‚ "!* œ œ œ % % !Þ!!!"!& "Þ!& ‚ "! "Þ!& ‚ "! "Þ!& ‚ "!% œ "& ‚ "!"$ œ "Þ& ‚ "!"% œ "" 

# "Î#

"Î#

52.

"#"

54.

$#B& 

"Î&

53.

#( Œ  "#&

55.

)"+% 

"Î$

œ ""

œ $#"Î& B& 

"Î&

œ #B

"Î%

$ "Î$

$ œ –Œ  — &

œ )""Î% +% 

44

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"Î%

œ

$ &

œ $k+k


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CHAPTER 0 REVIEW ˆ",!!!B' ‰"Î$ œ ",!!!"Î$ ˆB' ‰"Î$

57.

ˆ#&B# ‰"Î# œ #&"Î# ˆB# ‰"Î# Ê not a real number

58.

B"# C# 

59.

B"# Œ % C

60.

-#Î$ -&Î$  -#Î$ 

61.

+"Î% +$Î%  +*Î# 

62.

'%#Î$ œ ˆ'%"Î$ ‰ œ %# œ "'

64.

"' Œ  )"

66.

) Œ  #(

67.

"' Œ  '#&

68.

#"'B$ 

56.

œ "!B#

œ B"# 

"Î#

"Î#

78. 79.

-(Î$ œ  #Î$  -

"Î#

"Î$

"Î#

œ ˆ-*Î$ ‰

"Î$

+*Î# œ  "Î% $Î%  + +

"Î#

#Î$

#Î$

œŒ

'#&  "'

$

"Î$

"Î#

$ È #( #( $ œ $ œ È"#& "#& &

œŒ

C%  B"#

"Î#

œ

C# B'

œ -

+*Î# œ  "Î#  +

63.

$#$Î& œ

65.

$# Œ  #%$

"Î#

œ ˆ+)Î# ‰

"Î#

œ ˆ+% ‰

"Î#

œ +#

" " " " œ $ œ $ œ "Î& $#$Î& # ) $# 

ˆ$#"Î& ‰ $##Î& ## % œ œ # œ # #Î& #%$ $ * #%$"Î&  #

#Î&

œ

ˆ#("Î$ ‰ #(#Î$ $# * œ #Î$ œ œ œ # # "Î$ ) # % )  #

ˆ'#&"Î% ‰ '#&$Î% &$ "#& œ œ $ œ $ $Î% "' # ) "'"Î%  $

$Î%

œ

œ #"'#Î$ B$ 

È$' œ ' Ê

œ ˆ-$ ‰

+*Î# œ  #Î%  +

ˆ"'"Î% ‰ "'$Î% #$ ) œ œ $ œ $ $Î% )" $ #( )""Î% 

#( œŒ  )

$Î%

76.

œ B' kCk

$

œ

#Î$

73.

"Î#

#

$Î%

70.

"Î$

C# 

#Î$

71.

74.

% % È ) % 7) È 8% 7# k8k % 7 8 œ Ë :"' œ È % :% :"'

œ $'B#

69.

:+Î# :+Î$ :$+Î' :#+Î' :&+Î' œ œ +Î' œ :%+Î' œ :#+Î$ +Î' +Î' : : :

È%* œ (

72.

ÈB# C% œ ÈB# ÈC% œ kBkC# 77.

Ê &

75.

Ê

È* * $ œ œ È#& #& &

$ È B$ œ B

& & È +"& , "! +"& È , "! +$ , # œ œ & È -& -&

È&!  È) œ È#&È#  È%È# œ &È#  #È# œ (È#

È"#  È$  È#( œ È%È$  È$  È*È$ œ #È$  È$  $È$ œ $È$  $È$ œ !

45

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CHAPTER 0 REVIEW 80. 81.

83.

$ $ $ $ $ $ $ $ $ È #%B%  È $B% œ È )B$ È $B  È B$ È $B œ #BÈ $B  BÈ $B œ BÈ $B

È( È&

œ

È( È& È$& † œ È& È& &

82.

$ $ $ È È " " È % % % œ † œ œ $ $ $ $ È È È È # ) # # %

84.

$ $ $ È # # & #È & #È & œ † œ œ $ $ $ $ È È È È & #& #& & "#&

85.

È#

87.

È#B

89.

3rd degree, binomial

90.

2nd degree, trinomial

91.

2nd degree, monomial

92.

4th degree, trinomial

93.

#B  $  $B  % œ #B  '  $B  "# œ &B  '

94. 95. 96. 97. 98. 99.

&

$

œ

È# È# # † œ È È & # & #

) ) È# )È # )È # œ † œ œ È) È) È# È"' % È œ# #

œ

86.

È#B È#B #B † œ È#B $ $È#B

88.

È& &

œ

È& È& & " † œ œ È È È & & & & &

$ $ $ $ $È (B $È (B È %*B# $È $%$B$ œ † $ œ $ È%*B# # # #È%*B# #"B œ $ È # %*B#

$B# B  "  #BB  $  B# B  # œ $B$  $B#  #B#  'B  B$  #B# œ #B$  (B#  'B $B  #$B  # œ *B#  'B  'B  % œ *B#  "#B  %

$B  C#B  $C œ 'B#  *BC  #BC  $C# œ 'B#  (BC  $C#

%+  #, #+  $,  œ )+#  "#+,  %+,  ', # œ )+#  )+,  ', #

D  $$D #  D  " œ $D$  D#  D  *D#  $D  $ œ $D$  "!D#  #D  $ +8  #+8  " œ +#8  +8  #+8  # œ +#8  +8  #

100. ŠÈ#  B‹ œ ŠÈ#  B‹ŠÈ#  B‹ œ ŠÈ#‹  BÈ#  BÈ#  B# œ #  #BÈ#  B# #

#

101. ŠÈ#  "‹ŠÈ$  "‹ œ È'  È#  È$  "

$ $ $ $ $ $ $ $ 102. ŠÈ $  #‹ŠÈ *  #È $  %‹ œ È #(  #È *  %È $  #È *  %È $  ) œ $  ) œ &

46

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CHAPTER 0 REVIEW È$  "

# ŠÈ $  " ‹

# ŠÈ $  " ‹

œ È$  "

È$  "

104.

#ŠÈ$  È#‹ #ŠÈ$  È#‹ È$  È# # # œ † œ œ # # È$  È# È$  È# È$  È# $# ŠÈ $ ‹  ŠÈ # ‹

#

œ

È$  " È$  "

# ŠÈ $  " ‹

103.

#

œ

ŠÈ $ ‹  " # #

œ

$"

œ

œ

#

#ŠÈ$  È#‹

" œ #ŠÈ$  È#‹

105.

ÈB  # #BˆÈB  #‰ #BˆÈB  #‰ #B #B œ † œ œ # ÈB  # ÈB  # ÈB  # B% ˆÈB‰  ##

106.

ÈB  ÈC

107.

ÈB  #

108.

109.

111.

ÈB  ÈC

&

œ

œ

ÈB#  ÈBC  ÈBC  C ÈB  ÈC ÈB  ÈC B  #ÈBC  C † œ œ # # ÈB  ÈC ÈB  ÈC BC ˆÈB‰  ˆÈC‰

ÈB  # ÈB  # ˆÈB‰#  ## B% † œ œ È ˆ È ‰ ˆ È & B# & B# & B  #‰

# "  È+ "  È+ "  È+ "#  ˆÈ+‰ "+ œ † œ œ + + "  È+ +ˆ"  È+‰ +ˆ"  È+‰

$B# C# C œ $ 'B C #B

B#  #B  " #B  $ #B$  (B#  )B  $ #B$  $B# %B#  )B %B#  'B #B  $ #B  $ !

110.

112.

%+# , $  '+, % %+# , $ '+, % œ  #, # #, # #, # œ #+# ,  $+, # B$  #B  $  B' # " B  " B&  !B%  B$  $B#  #B  $ B&  B$ #B$  $B#  #B #B$  #B  $B# $  $B# $ ' #

113. $>$  $> œ $>>#  " œ $>>  ">  "

114. &<$  & œ &<$  " œ &<$  "$  œ &<  "<#  <  "

115. 'B#  (B  #% œ $B  )#B  $

116. $+#  +B  $+  B œ +$+  B  "$+  B œ $+  B+  " 47

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CHAPTER 0 REVIEW 117. )B$  "#& œ #B$  &$ œ #B  &#B#  #B&  &# ‘ œ #B  &%B#  "!B  #&

118. 'B#  #!B  "' œ #$B#  "!B  ) œ #$B  #B  %

119. B#  'B  *  ># œ B  $B  $  ># œ B  $#  ># œ B  $  >B  $  > 120. $B#  "  &B œ $B#  &B  " Ê prime

121. )D $  $%$ œ #D $  ($ œ #D  (#D #  #D (  (# ‘ œ #D  (%D#  "%D  %* 122. "  "%,  %*, # œ %*, #  "%,  " œ (,  "(,  " œ (,  "#

123. "#"D #  %  %%D œ "#"D #  %%D  % œ ""D  #""D  # œ ""D  ##

124. '%C$  ",!!! œ ))C$  "#& œ )#C$  &$ ‘ œ )#C  &%C#  "!C  #&

125. #BC  %DB  AC  #DA œ #BC  #D  AC  #D œ C  #D#B  A 126. B)  B%  " œ B)  #B%  "  B% œ ˆB%  "‰ˆB%  "‰  B%

# # œ ˆB%  "‰  ˆB# ‰ œ ˆB%  "  B# ‰ˆB%  "  B# ‰

œ ˆB%  #B#  "  B# ‰ˆB%  B#  "‰ œ ˆB#  "‰ˆB#  "‰  B# ‘ˆB%  B#  "‰

œ ˆB#  "  B‰ˆB#  "  B‰ˆB%  B#  "‰

127.

129.

130.

131.

132.

B#

#B B  # " œ œ  %B  % B# B  #B  #

128.

+#

+#  * +$ +  $+  $ œ œ  '+  * +$ +  $+  $

B#  %B  % B#  &B  ' B  #B  # B  #B  $ † œ † œ B  #B  $ B# B# B# B# #C#  ""C  "& C#  #C  ) #C  &C  $ C  %C  # #C  & † # œ † œ C#  'C  ) C C' C  %C  # C  $C  # C# #>#  >  $ "!>  "& #>#  >  $ $>#  >  % ƒ œ † $>#  (>  % $>#  >  % $>#  (>  % "!>  "& #>  $>  " $>  %>  " >" œ † œ $>  %>  " &#>  $ &

:#  (:  "# :#  * :#  (:  "# :# :  $:  % :# ƒ œ † œ † :$  ):#  %: :# :$  ):#  %: :#  * ::#  ):  % :  $:  $ ::  % œ # :  ):  %:  $ 48

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CHAPTER 0 REVIEW 133.

B#  B  ' B#  B  ' B#  % † # ƒ # # B  B  ' B  B  # B  &B  ' B#  B  ' B#  B  ' B#  &B  ' œ # † † B  B  ' B#  B  # B#  % B  $B  # B  $B  # B  #B  $ B  $B  #B  $ œ œ † † B  $B  # B  #B  " B  #B  # B  ## B  "

134. Œ

135.

136.

137.

138.

#B  ' #B#  #B  % B#  B  # #B  ' B#  #& B#  B  # ƒ œ † † #  # # # B& B  #& B  #B  "& B  & #B  B  # B  #B  "& #B  $ B  &B  & B  #B  " œ † † œ" B& #B  #B  " B  &B  $

# $B #B  & $BB  % #B  "! $B#  "#B  œ  œ  B% B& B  %B  & B  &B  % B  %B  & B  %B  & $B#  "!B  "! œ B  %B  & &B $B  ( #B  " &B B  ' &BB  # B  'B  #   œ  œ  B# B# B# B# B# B  #B  # B  #B  # &B#  "!B B#  %B  "# œ  B  #B  # B  #B  # %B#  'B  "# ##B#  $B  ' œ œ B  #B  # B  #B  #

B B B   B" B# B$ BB  #B  $ BB  "B  $ BB  "B  # œ   B  "B  #B  $ B  #B  "B  $ B  $B  "B  # B$  &B#  'B B$  %B#  $B B$  $B#  #B œ   B  "B  #B  $ B  "B  #B  $ B  "B  #B  $ $B$  "#B#  ""B B$B#  "#B  "" œ œ B  "B  #B  $ B  "B  #B  $ B $B  ( #B  " B $B  ( #B  "   œ   B" B# B# B" B# B# B B  ' œ  B" B# BB  # B  'B  " œ  B  "B  # B  #B  " # B  #B B#  (B  ' &B  ' œ  œ B  "B  # B  "B  # B  "B  #

49

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CHAPTER 0 REVIEW 139.

140.

141.

143.

144.

$B  " &B#  $ B $BB  "B  " &B#  $B  " BB#    œ   # # # # B B B" B B  " B B  " B B  " $B$  'B#  $B &B$  &B#  "&B  "& B$ œ   B# B  " B# B  " B# B  " $ # B  B  "#B  "& œ B# B  " $B B#  %B  $ B#  B  ' $B B  $B  " B  $B  #  #  œ   # B  " B  $B  # B % B  " B  "B  # B  #B  # $B B$ B$ $B œ   œ B" B# B# B"

&B # $B# ) " B



œ

" C

BC

&B $B# &B ) #! ƒ œ † # œ # ) # $B $B

œ

BCŠ B"  C" ‹ BCB  C

B"  C" œ C"  B"

" B " C

 

" C " B

Chapter 0 Test

œ

œ

142.

BCˆ B" ‰  BCŠ C" ‹ BCB  C

BCŠ B"  C" ‹ BCŠ "C



" B‹

œ

œ

$B C 'B C#

œ

$B 'B $B C# C ƒ # œ † œ C C C 'B #

œ

CB BC

CB BCB  C

BCˆ B" ‰  BCŠ C" ‹ BCŠ "C ‹



BCˆ B" ‰

(page 82)

1.

odd integers: (, ", $

2.

prime numbers: $

3.

commutative property of addition

4.

distributive property

5.

% ! B Ÿ # Ê

6.

Ð∞, $Ñ ∪ Ò', ∞Ñ Ê

oqÐïïÓqp %

íïÑqqÒïî

#

$

7.

Since "( ! !, k"(k œ "( œ "(

8.

9.

distance œ k"#  %k œ k"'k œ "'

10.

distance œ k"#  #!k œ k)k œ )

11.

B% B& B# œ B%&# œ B""

12.

<# <$ = <& = < œ œ < % =# < % =# =

14.

Œ

16.

!Þ!!!$%& œ $Þ%& ‚ "!%

+" +#  +$

#

13. 15.

+"  +$

#

œ

%&!,!!! œ %Þ& ‚ "!&

œ

+# œ+ +$

If B ! !, then B  ( ! !. Then kB  (k œ B  (.

B! B# B# % ' #% œ  Œ  œ ˆB ‰ œ B # # B B '

50

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'

'


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CHAPTER 0 TEST 17.

$Þ( ‚ "!$ œ $,(!!

19.

#&+% 

21.

)>' Œ  #(=*

"Î#

œ #&"Î# +% 

"Î#

#Î$

22.

24.

25.

26.

œŒ

œ &+#

#(=*  )>'

#Î$

$ $ $ È #(+' œ È #(È +' œ $+#

#(#Î$ =* 

#Î$

œ

)#Î$ >' #Î$

œ

18.

"Þ# ‚ "!$ œ !Þ!!"#

20.

Œ

$'  )"

ˆ#("Î$ ‰# =' # )"Î$  >%

23.

œ

ˆ$'"Î# ‰ $'$Î# #"' œ $ œ $Î# "Î# )" (#* )"  $

$Î#

œ *=' %>%

È"#  È#( œ È%È$  È*È$

œ #È $  $È $ œ &È $

$ $ $ $ $ $ $ $ #È$B%  $BÈ #%B œ #ÈB$ È $B  $BÈ )È $B œ #BÈ $B  $B#È $B œ #BÈ $B  'BÈ $B $ œ %BÈ$B $

$

ÈB  # BˆÈB  #‰ BˆÈB  #‰ B B œ † œ œ # ÈB  # ÈB  # ÈB  # B% ˆÈB‰  ##

ÈB  ÈC ÈB  ÈC

œ

ÈB  ÈC ÈB  ÈC ˆÈB‰#  ˆÈC‰# BC † œ œ ÈB  ÈC ÈB  ÈC ÈB#  ÈBC  ÈBC  ÈC# B  #ÈBC  C

ˆ+#  $‰  ˆ#+#  %‰ œ +#  $  #+#  %

28.

$+$ , # #+$ , %  œ '+' , '

29.

$B  %#B  ( œ 'B#  #"B  )B  #) œ 'B#  "$B  #)

30.

+8  #+8  $ œ +#8  $+8  #+8  ' œ +#8  +8  '

31.

B#  %B#  % œ B%  %B#  %B#  "' œ B%  "'

27.

œ +#  (

32. 33.

35. 37.

B#  B  ##B  $ œ #B$  $B#  #B#  $B  %B  ' œ #B$  &B#  (B  ' $% 'B  "*  B$ B  $ 'B#  B  #$ 'B#  ")B "*B  #$ "*B  &( $%

$B  'C œ $B  #C

"!>#  "*>A  'A# œ &>  #A#>  $A

34.

36. 38.

B#  #B  " #B  " #B$  $B#  !B  " #B$  B# %B#  !B %B#  #B #B  " #B  " !

B#  "!! œ B#  "!# œ B  "!B  "! $+$  '%) œ $ˆ+$  #"'‰ œ $+  'ˆ+#  '+  $'‰

51

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SECTION 1.1 39.

B%  B#  "# œ ˆB#  %‰ˆB#  $‰ œ B  #B  #ˆB#  $‰

41.

B # B#  œ œ" B# B# B#

42.

43.

44.

45.

46.

40.

'B%  ""B#  "! œ $B#  ##B#  &

B B BB  " BB  " B#  B  B#  B #B  œ  œ œ B" B" B  "B  " B  "B  " B  "B  " B  "B  "

B#  B  #! B#  #& B  &B  % B  &B  & B  &# † œ † œ B#  "' B& B  %B  % B& B% B# " +

B# B#  % B# B" " ƒ œ † œ  #B  " B" B  "B  " B  #B  # B  "B  #



" ,

" ,

œ

+, ˆ +"  ", ‰ +, ˆ +" ‰  +, ˆ ", ‰ ,+ œ œ "‰ ˆ + + +, ,

B" œ B"  C"

" B " B

Exercises 1.1



" C

œ

BCˆ B" ‰

BCŠ B"



" C‹

œ

BCˆ B" ‰

C 

BCŠ C" ‹

œ

C CB

(page 93)

1.

root, solution

2.

identity

3.

no

4.

conditional

5.

linear

6.

rational

7.

one

8.

denominator

9.

B$ œ " no restrictions

10.

" B  ( œ "% # no restrictions

11.

13.

15.

) & œ B$ B# B$ Á ! B# Á ! BÁ$ B Á # B Á $, B Á # " &B œ # B$ B  "' " &B œ B  %B  % B$ B$ Á ! B% Á ! B% Á ! BÁ$ B Á % BÁ% B Á $, B Á %, B Á %

14.

16.

" œ "# B BÁ!

12.

$ œ *B B# BÁ#

B % œ B$ B% B$ Á ! B% Á ! BÁ$ B Á % B Á $, B Á % " & œ # B#  $B  % B " & œ # B  "B  % B B" Á ! B% Á ! BÁ ! B Á " BÁ% B Á ", B Á %, B Á !

52

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SECTION 1.1 17.

20.

23.

25.

#B  & œ "& #B  &  & œ "&  & #B œ "! #B "! œ # # Bœ& conditional equation

18.

$7  # œ #7  $  7 21. $7  ' œ #7  '  7 $7  ' œ $7  ' identity

#+  " œ $+  #  + #+  # œ $+  '  + #+  # œ #+  ' #+  #+  # œ #+  #+  ' # Á ' contradiction $B  $ œ

'B  ") # 'B  ") $B  * œ # 'B  ") #$B  * œ # † # 'B  ") œ 'B  ") identity

$B  # œ B  ) $B  B  # œ B  B  ) #B  # œ ) #B  #  # œ )  # #B œ ' #B ' œ # # Bœ$ conditional equation B( œ( # B( #† œ #( # B  ( œ "% B  (  ( œ "%  ( Bœ( conditional equation

24.

26.

19.

22.

#8  #  & œ #8 #8  %  & œ #8 #8  " œ #8 #8  #8  " œ #8  #8 " Á ! contradiction

B  ( œ "% # B  (  ( œ "%  ( # B œ #" # B # † œ ##" # B œ %# conditional equation

B# œ B  %B  %  "' B# œ B#  "'  "' B# œ B# identity

BB  # œ B  "# B#  #B œ B  "B  " B#  #B œ B#  #B  " B#  B#  #B œ B#  B#  #B  " #B œ #B  " #B  #B œ #B  #B  " !Á" contradiction

53

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SECTION 1.1 27.

29.

31.

33.

35.

$ œ" ,$ $ ,  $ † œ ,  $" ,$ $œ,$ $$œ,$$ 'œ, conditional equation

#B#  &B  $ œ #B  "B  $ #B#  &B  $ œ #B#  &B  $ identity

#B  ( œ "!  B $B  ( œ "! $B œ $ Bœ" &B  # œ #B  % &B  "! œ #B  ) $B  "! œ ) $B œ ") Bœ' (#B  &  'B  ) œ ( "%B  $&  'B  %) œ ( )B  "$ œ ( )B œ #! #! & Bœ œ ) #

28.

30.

32.

34.

36.

B#  )B  "& œ B  $B  & B#  )B  "& œ B#  #B  "& B#  B#  )B  "& œ B#  B#  #B  "& )B  "& œ #B  "& )B  )B  "& œ #B  )B  "& "& œ "!B  "& "&  "& œ "!B  "&  "& $! œ "!B $! "!B œ "! "! $œB conditional equation #B#  &B  $ œ #BŒB 

"*  # #B#  &B  $ œ #B#  "*B #B#  #B#  &B  $ œ #B#  #B#  "*B &B  $ œ "*B &B  &B  $ œ "*B  &B $ œ "%B $ "%B œ "% "% $  œB "% conditional equation *+  $ œ "&  $+ '+  $ œ "& '+ œ ") +œ$ &<  % œ &<  % &<  #! œ &<  #! "!<  #! œ #! "!< œ %! <œ% 'B  &  %B  # œ " 'B  $!  %B  ) œ " #B  $) œ " #B œ $( $( Bœ #

54

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SECTION 1.1 37.

40.

43.

45.

& D)œ( $ & D œ "& $ & $ † D œ $"& $ &D œ %& Dœ* $:  : œ % ( $: (Œ  : œ (% ( $:  (: œ #) %: œ #) :œ(

38.

41.

$B  " " œ #! # $B  " " #! † œ #! † #! # $B  " œ "! $B œ * Bœ$

$B B(  œ %B  " $ # $B B( 'Œ   œ '%B  " $ # #$  B  $B  ( œ #%B  ' '  #B  $B  #" œ #%B  ' &B  #( œ #%B  ' "*B  #( œ ' "*B œ #" #" Bœ "*

% C  "# œ % $ % C œ "' $ % $ † C œ $"' $ %C œ %) C œ "#

39.

$B  # ( œ #B  $ $ $B  # ( $† œ $Œ#B   $ $ $B  # œ 'B  ( $B  # œ ( $B œ * B œ $ 44.

46.

42.

D #œ% & D œ# & D & † œ &# & D œ "!

( "& B& œ B # #

( "& #Œ B  & œ #ŒB   # # (B  "! œ #B  "& &B  "! œ "& &B œ & Bœ"

( B %B  $  œ ' ' ' ( B %B  $ 'Œ#B    œ ' † ' ' ' "#B  (  B œ %B  $ "$B  ( œ %B  $ *B  ( œ $ *B œ "! "! Bœ * #B 

$B $%  B œ # # $B $%  B #”##B  "  •œ#† # # %#B  "  $B œ $%  B )B  %  $B œ "#  $B &B  % œ "#  $B )B  % œ "# )B œ "' B œ # ##B  " 

55

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SECTION 1.1 47.

$ $B  #  "!B  % œ ! #

48.

$ #” $B  #  "!B  %• œ #! # $$B  #  #!B  ) œ ! *B  '  #!B  ) œ ! ""B  "% œ ! ""B œ "% "% Bœ "" 49.

C  ## C# œC# $ $ C  ## C# $– œ $ C  #  Œ  — $ $

++  $  & œ +  "+  " +#  $+  & œ +#  #+  " $+  & œ #+  " &œ+" %œ+

50.

C  ## œ $C  '  C# C#  %C  % œ C#  $C  ' %C  % œ $C  ' C%œ' Cœ#

51.

53.

BB  # œ B  "#  " B#  #B œ B  "B  "  " B#  #B œ B#  #B  "  " B#  #B œ B#  #B ! œ ! Ê identity

++  $  & +  "# œ 7 7 ++  $  & +  "# 7” œ 7 • – — 7 7

52.

>  ">  " œ >  #>  $  % > #  " œ >#  >  '  % " œ >  # >  " œ # > œ "

B  #B  $ œ B  $B  % B#  &B  ' œ B#  (B  "# &B  ' œ (B  "# "#B  ' œ "# "#B œ ' ' " Bœ œ "# #

#=  #  =  $# œ ==  &  #Œ

"(  = # #=  %  =#  '=  * œ =#  &=  "(  #= =#  )=  "$ œ =#  (=  "( )=  "$ œ (=  "( =œ%

56

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SECTION 1.1 54.

56.

58.

59.

$ " %  œ B # B $ " % #BŒ   œ #B † B # B 'Bœ) Bœ#

$ " $  œ B# B B# $ " $ BB  #Œ   œ BB  # † B# B B# $B  "B  # œ $B $B  B  # œ $B %B  # œ $B Bœ# The answer does not check. Ê no solution

55.

57.

# " "  œ B" $ B" # " " $B  "Œ   œ $B  " † B" $ B" '  "B  " œ $" 'B" œ $ B( œ $ B œ % *>  ' ( œ >$ >>  $ *>  ' ( >>  $” • œ >>  $ † >>  $ >$ *>  ' œ (> #>  ' œ ! #> œ ' > œ $ The answer does not check. Ê no solution

##B  " $B# œ $B  & $B  & ##B  " $B# $B  &”B  • œ $B  & † $B  & $B  & # B$B  &  ##B  " œ $B $B#  &B  %B  # œ $B# B# œ ! B œ # B

# % œ +  (+  # +  $+  # # % +  (+  #+  $ † œ +  (+  #+  $ † +  (+  # +  $+  # #+  $ œ %+  ( #+  ' œ %+  #) #+ œ $% + œ "(

57

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SECTION 1.1 60.

61.

62.

# " "  œ # 8# 8" 8 8# # " "  œ 8  #8  " 8# 8" # " " 8  #8  "Œ   œ 8  #8  " † 8  #8  " 8# 8" #8  "  "8  # œ " #8  #  8  # œ " $8 œ " " 8œ $ #B  $ $B  # &B  #  # œ #  &B  ' B  B  ' B % #B  $ $B  # &B  #  œ B  $B  # B  $B  # B  #B  # B  ##B  $  B  #$B  # œ B  $&B  # #B#  B  '  $B#  %B  % œ &B#  "$B  ' &B#  $B  "! œ &B#  "$B  ' $B  "! œ "$B  ' "!B œ % % # Bœ œ "! & B#

emultiply by common denominatorf

$B #B B#  œ # B#  B B#  &B B  'B  & $B #B B#  œ B  &B  " BB  " BB  & $ # B#  œ B" B& B  &B  " emultiply by common denominatorf $B  &  #B  " œ B  # $B  "&  #B  # œ B  # B  "$ œ B  # "$ Á # Ê no solution

58

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SECTION 1.1 63.

$B  & $ #$B  #  # œ $ B ) B % B  #B#  #B  % $B  & $ #$B  #  œ B  #B#  #B  % B  #B  # B  #B#  #B  % B  #$B  &  ˆB#  #B  %‰$ œ #B  #$B  # emultiply by common denominatorf $B#  B  "!  $B#  'B  "# œ 'B#  )B  ) 'B#  (B  # œ 'B#  )B  ) "&B œ "! "! # Bœ œ "& $

64.

65.

66.

" $8  % "  œ 8  ) &8#  %#8  "' &8  # " $8  % "  œ 8  ) &8  #8  ) &8  # &8  #"  $8  % œ 8  ) emultiply by common denominatorf &8  #  $8  % œ 8  ) #8  ' œ 8  ) 8œ# " #$8  " "  œ ""  8 (8#  (%8  $$ (8  $ " #$8  " "  # œ 8  "" (8  (%8  $$ (8  $ " '8  # "  œ 8  "" (8  $8  "" (8  $ (8  $  '8  # œ 8  """ (8  $  '8  # œ 8  "" 8  & œ 8  "" #8 œ ' 8œ$

emultiply by common denominatorf

% # "  # œ #  "$+  %) +  ")+  $# + +' % # "  œ +  "'+  $ +  "'+  # +  $+  # %+  #  #+  $ œ "+  "' emultiply by common denominatorf %+  )  #+  ' œ +  "' #+  "% œ +  "' + œ # +#

59

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SECTION 1.1 67.

68.

69.

70.

& # ' "  œ  C% C# C  # C#  'C  ) & % " œ  C% C  # C  #C  % &C  # œ %C  %  " emultiply by common denominatorf &C  "! œ %C  "'  " &C  "! œ %C  "& Cœ& ' $ "  œ # #+  ' $  $+ +  %+  $ ' $ "  œ #+  $ $"  + +  $+  " $ " "  œ +$ +" +  $+  " $+  "  "+  $ œ " emultiply by common denominatorf $+  $  +  $ œ " %+  ' œ " %+ œ ( ( +œ % $C #C )  œ '  $C #C  % %  C# $C #C )  œ $#  C #C  # #  C#  C C C )  œ #C #C #  C#  C emultiply by common denominatorf C#  C  C#  C œ ) #C  C#  #C  C# œ ) %C œ ) C œ # Ê The solution does not check, so the equation has no solution. $  #+ #  $+ &+  #  œ # +#  '  &+ +#  '  + + % #+  $ $+  # &+  #  œ +  #+  $ +  $+  # +  #+  # +  ##+  $  +  #$+  # œ +  $&+  # #+#  +  '  $+#  %+  % œ &+#  "$+  ' &+#  $+  "! œ &+#  "$+  ' "!+ œ % % # +œ œ "! &

emultiply by common denominatorf

60

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SECTION 1.1 71.

+ $+  # "œ # +# +  %+  % + " $+  #  œ +  #+  # +# " + $+  # +  #+  #”  "• œ +  #+  # † ” • +  #+  # +# ++  #  +  #+  # œ $+  # +#  #+  ˆ+#  %+  %‰ œ $+  # +#  #+  +#  %+  % œ $+  # #+  % œ $+  # +œ#

72.

73.

76.

B" B# "  #B  œ B$ B$ $B B" B# #B  "  œ B$ B$ B$ B  $B  "  B  $B  # œ B  $#B  " B#  %B  $  B#  B  ' œ #B#  &B  $ #B#  $B  $ œ #B#  &B  $ )B œ ! ! Bœ œ! ) 5 œ #Þ#: 5 #Þ#: œ #Þ# #Þ# 5 œ: #Þ# " # 1< 2 $ " œ $ † 1 <# 2 $ œ 1 <# 2 1 <# 2 œ 1 <#

Z œ $Z

$Z $Z 1 <# $Z œ2 1 <#

74.

77.

+B  , œ ! +B œ , , Bœ +

" # 1< 2 $ " œ $ † 1 <# 2 $ œ 1 <# 2 1 <# 2 œ 12

Z œ $Z

emultiply by common denominatorf

75.

78.

$Z $Z 12 $Z œ <# 12

61

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: œ #6  #A :  #6 œ #A :  #6 #A œ # # :  #6 œA # B. 5 B. D5 œ †5 5 D5 œ B  . .  D5 œ B . œ B  D5 Dœ


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SECTION 1.1 T8 œ P 

79.

T8  P œ

0 T8  P œ =3 0 T8  P =3 œ 3 3 0 T8  P œ= 3

85.

87.

" 0 " 0 :; † 0 :; :; :; ;: :; ;:

T8 œ P 

80.

=3 0

0 T8  P œ 0 †

82.

=3 0

T8  P œ

" "  : ; " " œ 0 :; Œ   : ; œ 0;  0: œ 0 ;  : 0 ;  : œ ;: œ

81.

=3 0

0 T8  P œ 0 †

=3 0

=3 0

J <#

J <# J <# Q1 J <# œ7 Q1

=3 0

0 T8  P œ =3 0 T8  P =3 œ T8  P T8  P =3 0œ T8  P 83.

B C  + , C , C ,† , C

œ"

84.

B + B œ , Š"  ‹ + B œ , Š"  ‹ +

œ"

œ0

" < " <<" <# † < <" <# <" <# <" <# <#  <" <" <# <#  <"

" "  <" <# " " œ <<" <# Œ   <" <# œ <<#  <<" œ <<#  <"  <<#  <"  œ <#  <" œ

86.

œ<

6 œ +  8  ". 6 œ +  8.  . 6  +  . œ 8. 6+. 8. œ . . 6+. œ8 .

88.

7Q 1 <# 7Q 1 # œ †< <# œ 7Q 1 7Q 1 œ Q1

J œ

B C  œ" + , B C +, Š  ‹ œ +, † " , + ,B  +C œ +, ,B œ +,  +C ,B œ +,  C ,B +,  C œ ,C ,C ,B œ+ ,C

" " " œ  < <" <# " " " <<" <# † œ <<" <# Œ   < <" <# <" <# œ <<#  <<" <" <#  <<" œ <<# <" <#  < œ <<# <" <#  < <<# œ <#  < <#  < <<# <" œ <#  <

6 œ +  8  ". 6  + œ 8  ". 6+ 8  ". œ 8" 8" 6+ œ. 8"

62

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SECTION 1.1 89.

+ œ 8  #

")! 8 ")! +8 œ 8  # †8 8 +8 œ 8  #")! +8 œ ")!8  $'! $'! œ ")!8  +8 $'! œ 8")!  + $'! œ8 ")!  + Vœ

91.

90.

+  6< "< +  6< W "  < œ "  < "< W "  < œ +  6< W  W<  6< œ + Wœ

" " <"



" <#



" <$

<" <# <$ "

<" <# <$ Š <""  <"#  <"$ ‹ <" <# <$ Vœ <# <$  <" <$  <" <# V<# <$  <" <$  <" <#  œ <" <# <$ V<# <$  V<" <$  V<" <# œ <" <# <$ V<" <$  V<" <#  <" <# <$ œ V<# <$ <" V<$  V<#  <# <$  œ V<# <$ V<# <$ <" œ V<$  V<#  <# <$ Vœ

92.

" " <"



" <#



" <$

<" <# <$ "

<" <# <$ Š <""  <"#  <"$ ‹ <" <# <$ Vœ <# <$  <" <$  <" <# V<# <$  <" <$  <" <#  œ <" <# <$ V<# <$  V<" <$  V<" <# œ <" <# <$ V<# <$  V<" <$  <" <# <$ œ V<" <# <$ V<#  V<"  <" <#  œ V<" <# V<" <# <$ œ V<#  V<"  <" <# Vœ

93.

Answers may vary.

95.

#&B 

# "Î#

# ‘"Î#

œ &B

œ &kBk

94.

Answers may vary.

96.

#&:# Œ  "'; %

"Î#

# "Î#

&: œ –Œ #  — %;

63

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œ

&k:k %; #


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SECTION 1.1

97.

Œ

"#&B$  )C'

#Î$

œŒ

)C'  "#&B$

#Î$

98.

)#Î$ C 

' #Î$

œ

99.

"#&#Î$ B$ #Î$

œ

%

"Î$

#("Î$ C$ 

"Î$

",!!!"Î$ B' "Î$ $C $C œ œ "!B# "!B# œ

$ $ 100. È "#&C* œ É &C$ $ œ &C$

+% , "# +, $ k+, $ k % œ œ Ë Œ  D) D# D#

Exercises 1.2

#(C$  ",!!!B'

%C% #&B#

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102. Ê

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(page "!")

1.

add

2.

perimeter

3.

amount

4.

break point

5.

rate, time

6.

!Þ!&$! œ "Þ&

7.

Let B œ the score on the first exam. Then B  & œ the score on the midterm, and B  "$ œ the score on the final. Sum of scores œ *! $ B  B  &  B  "$ œ *! $ $B  ") œ *! $ $B  ") œ #(! $B œ #&# B œ )% His score on the first exam was 84.

8.

Let B œ the score on the first exam. Then her scores on the following tests were B  $, B  ' and B  *. Sum of scores œ '*Þ& % BB$B'B* œ '*Þ& % %B  ") œ '*Þ& % %B  ") œ #() %B œ #'! B œ '& Her score on the first exam was 65%.

9.

Let B œ the program development score. Sum of scores œ )' % )#  *!  B  () œ )' % B  #&! œ )' % B  #&! œ $%% B œ *% The program development score was 94.

10.

Let B œ the score on the final round. Sum of scores œ (# % ('  ')  (!  B œ (# % B  #"% œ (# % B  #"% œ #)) B œ (% She needs to shoot 74 on the final round.

64

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SECTION 1.2 11.

Let B œ the number of locks replaced. %!  #) †

Number of locks

12.

œ #$'

Let B œ the number of interviews. #!  !Þ(& †

%!  #)B œ #$' #)B œ "*' Bœ( 7 locks can be changed for $236.

Number of interviews

œ &'

#!  !Þ(&B œ &' !Þ(&B œ $' B œ %) He interviewed 48 people.

13.

Let B œ the width. Then B  #' œ the height. Perimeter œ *# #B  #B  #' œ *# #B  #B  &# œ *# %B œ %! B œ "! The dimensions are 10 ft by 36 ft.

15.

Perimeter œ "% B  B  #  B  B  # œ "% %B  % œ "% %B œ "! & " " B œ œ # Ê The width is 2 feet. # # #

16.

Total Fence Length œ # † Square Fence Length B  B  #%  B  B  #% œ # † B  B  B  B %B  %) œ )B %) œ %B B œ "# The total fencing required is %B  %) œ %"#  %) œ *' feet.

17.

Total Area œ # † Triangular Area " " #!B  "'#! œ # † "'#! # # #!B  "'! œ $#! #!B œ "'! Bœ) The dimensions are 8 feet by 20 feet.

14.

18.

Let B œ the width. Then B  ""& œ the length. Perimeter œ &(! #B  #B  ""& œ &(! #B  #B  #$! œ &(! %B œ $%! B œ )& The dimensions are 85 ft by 200 ft.

Sum of angles œ ")! B  B  $!  B  $! œ ")! $B  '! œ ")! $B œ "#! B œ %! The angles measure %!°, (!° and (!°.

65

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SECTION 1.2 19.

20.

New Area

œ

Old Area

 !Þ&! †

Old Area

"#B  "!  "#B œ "#B  "!  !Þ&! † "#B  "! "#B  "#!  "#B œ "#B  "#!  'B  '! #%B  "#! œ ")B  ")! 'B œ '! B œ "! Ê The length of the living room is B  "! œ #! feet. Area œ &%

" . "#  ) œ &% # "!. œ &% . œ &Þ% Ê The depth is 5.4 inches

21.

Let B œ the amount invested at 7%. Then ##!!!  B œ the amount invested at 6%. Interest at 7%  Interest at 6% œ Total interest !Þ!(B  !Þ!'##!!!  B œ "%#! !Þ!(B  "$#!  !Þ!'B œ "%#! !Þ!"B œ "!! B œ "!!!! $10,000 was invested at 7% and $12,000 was invested at 6%.

22.

Let B œ the amount invested at 7%. Interest at 7%  Interest at 9% œ Total interest !Þ!(B  !Þ!*#!!!! œ &!!! !Þ!(B  ")!! œ &!!! !Þ!(B œ $#!! B ¸ %&("%Þ#* She needs to invest $45,714.29 at 7% to reach her goal.

23.

Let B œ the amount invested at each rate. Interest at 6%  Interest at 7%  Interest at 8% œ Total interest !Þ!'B  !Þ!(B  !Þ!)B œ #!$( !Þ#"B œ #!$( B œ *(!! $9,700 was invested at each rate, for a total investment of $#*,100.

66

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SECTION 1.2 24.

Let B œ the amount invested at 8%. Then $(,!!!  B œ the amount invested at 9 "# %. Interest at 9 12 % œ Interest at 8%  %&#Þ&!

!Þ!*&$(,!!!  B œ !Þ!)B  %&#Þ&! $&"&  !Þ!*&B œ !Þ!)B  %&#Þ&! $!'#Þ&! œ !Þ"(&B "(&!! œ B $17,500 is invested at 8% and $19,500 is invested at 9 12 %. 25.

Let B œ the number of full-price tickets sold. Then &)&  B œ the number of student tickets sold. #Þ&! †

26.

# of full-price

œ "#"(Þ#&

#Þ&!B  "Þ(&&)&  B œ "#"(Þ#& #Þ&!B  "!#$Þ(&  "Þ(&B œ "#"(Þ#& !Þ(&B œ "*$Þ&! B œ #&) Ê There were 327 student tickets sold.

Let B œ the cost of a student ticket. Cost of full-price

27.

# of student

 "Þ(& †

# of full-price



Cost of student

œ %*'!

%)!(  B)!!  %)! œ %*'! $$'!  $#!B œ %*'! $#!B œ "'!! B œ & Ê A student ticket cost $5.

Let : œ the original price. Original price

# of student

 Discount œ

28. New price

Wholesale cost

:  !Þ#!: œ '$Þ*' !Þ)!: œ '$Þ*' : œ (*Þ*& The original price was $79.95. 29.

Let B œ # of plates for equal costs. Cost of 1st machine

œ

Let A œ the wholesale cost.  Markup œ

Selling price

A  !Þ$!A œ &))Þ*! "Þ$!A œ &))Þ*! A œ %&$ The wholesale cost is $453. 30.

Cost of 2nd machine

Let B œ # of fasteners for equal costs. Cost of 1st machine

'!!  $B œ )!!  #B B œ #!! The break point is 200 plates.

œ

Cost of 2nd machine

"#!!  !Þ!!&B œ "&!!  !Þ!!"&B !Þ!!$&B œ $!! B ¸ )&("% The break point is about 85,714 fasteners.

67

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SECTION 1.2 31.

Let B œ # of computers to break even. Income œ Expenses "#(&B œ )*#&  )&!B %#&B œ )*#& B œ #" 21 computers need to be sold to break even.

32.

Let B œ # of meals to break even. Income œ Expenses 'B œ "$(Þ&!  %Þ(&B "Þ#&B œ "$(Þ&! B œ ""! More than 110 meals need to be sold to make a profit.

33.

Let B œ days for both working together.

34.

Let B œ hours for both working together.

Man in 1 day



Roofer in 1 day

œ

Total in 1 day

Crew 1 in 1 hour

Let B œ hours for both working together. Woman in 1 hour



Man in 1 hour

œ

Crew 2 in 1 hour

œ

Total in 1 hour

" " "  œ ) "! B " " " %!BŒ   œ %!BŒ  ) "! B &B  %B œ %! *B œ %! %! % Bœ œ% * * They can seal the parking lot in 4 49 hours.

" " "  œ ( % B " " " #)BŒ   œ #)BŒ  ( % B %B  (B œ #) ""B œ #) #) ' Bœ œ# "" "" 6 They can roof the house in 2 11 days. 35.



36.

Total in 1 hour

Let B œ days for both hoses to fill the pool. 1st hose in 1 day

" " "  œ # % B " " " %BŒ   œ %BŒ  # % B #B  B œ % $B œ % % " Bœ œ" $ $ They can mow the lawn in 1 "$ hours.



2nd hose in 1 day

œ

Total in 1 day

" " "  œ $ # B " " " 'BŒ   œ 'BŒ  $ # B #B  $B œ ' &B œ ' ' " Bœ œ" & & The pool can be filled in 1 15 days.

68

Full file at https://testbankuniv.eu/College-Algebra-11th-Edition-Gustafson-Solutions-Manual

College Algebra 11th Edition Gustafson Solutions Manual  

Full file at https://testbankuniv.eu/College-Algebra-11th-Edition-Gustafson-Solutions-Manual

College Algebra 11th Edition Gustafson Solutions Manual  

Full file at https://testbankuniv.eu/College-Algebra-11th-Edition-Gustafson-Solutions-Manual

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