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# Exercise Supplement

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. This module contains the exercise supplement for the chapter "Basic Properties of Real Numbers".

## Exercise Supplement

### Symbols and Notations ((Reference))

For the following problems, simplify the expressions.

#### Exercise 1

12+7( 4+3 ) 12+7( 4+3 )

61

#### Exercise 2

9( 42 )+6( 8+2 )3( 1+4 ) 9( 42 )+6( 8+2 )3( 1+4 )

#### Exercise 3

6[ 1+8( 7+2 ) ] 6[ 1+8( 7+2 ) ]

438

26÷210 26÷210

#### Exercise 5

( 4+17+1 )+4 141 ( 4+17+1 )+4 141

2

51÷3÷7 51÷3÷7

#### Exercise 7

( 4+5 )( 4+6 )( 4+7 ) ( 4+5 )( 4+6 )( 4+7 )

79

#### Exercise 8

8( 212÷13 )+2511[ 1+4( 1+2 ) ] 8( 212÷13 )+2511[ 1+4( 1+2 ) ]

#### Exercise 9

3 4 + 1 12 ( 3 4 1 2 ) 3 4 + 1 12 ( 3 4 1 2 )

37 48 37 48

#### Exercise 10

483[ 1+17 6 ] 483[ 1+17 6 ]

#### Exercise 11

29+11 61 29+11 61

8

#### Exercise 12

88 11 + 99 9 +1 54 9 22 11 88 11 + 99 9 +1 54 9 22 11

#### Exercise 13

86 2 + 99 3 104 5 86 2 + 99 3 104 5

##### Solution

43

For the following problems, write the appropriate relation symbol ( =,<,> ) ( =,<,> ) in place of the .

226 226

#### Exercise 15

9[ 4+3( 8 ) ]6[ 1+8( 5 ) ] 9[ 4+3( 8 ) ]6[ 1+8( 5 ) ]

252>246 252>246

#### Exercise 16

3( 1.06+2.11 )4( 11.019.06 ) 3( 1.06+2.11 )4( 11.019.06 )

#### Exercise 17

20 20

##### Solution

2>0 2>0

For the following problems, state whether the letters or symbols are the same or different.

> and ≮

different

#### Exercise 20

a=bandb=a a=bandb=a

#### Exercise 21

Represent the sum of c c and d d two different ways.

##### Solution

c+d;d+c c+d;d+c

For the following problems, use algebraic notataion.

8 plus 9

#### Exercise 23

62 divided by f f

##### Solution

62 f or62÷f 62 f or62÷f

#### Exercise 24

8 times ( x+4 ) ( x+4 )

#### Exercise 25

6 times x x , minus 2

6x2 6x2

#### Exercise 26

x+1 x+1 divided by x3 x3

#### Exercise 27

y+11 y+11 divided by y+10 y+10 , minus 12

##### Solution

( y+11 )÷( y+10 )12or y+11 y+10 12 ( y+11 )÷( y+10 )12or y+11 y+10 12

#### Exercise 28

zero minus a a times b b

### The Real Number Line and the Real Numbers ((Reference))

#### Exercise 29

Is every natural number a whole number?

yes

#### Exercise 30

Is every rational number a real number?

For the following problems, locate the numbers on a number line by placing a point at their (approximate) position.

2

3.6 3.6

1 3 8 1 3 8

0

4 1 2 4 1 2

#### Exercise 36

Draw a number line that extends from 10 to 20. Place a point at all odd integers.

#### Exercise 37

Draw a number line that extends from 10 10 to 10 10 . Place a point at all negative odd integers and at all even positive integers.

#### Exercise 38

Draw a number line that extends from 5 5 to 10 10 . Place a point at all integers that are greater then or equal to 2 2 but strictly less than 5.

#### Exercise 39

Draw a number line that extends from 10 10 to 10 10 . Place a point at all real numbers that are strictly greater than 8 8 but less than or equal to 7.

#### Exercise 40

Draw a number line that extends from 10 10 to 10 10 . Place a point at all real numbers between and including 6 6 and 4.

For the following problems, write the appropriate relation symbol ( =,<,> ). ( =,<,> ).

3 0 3 0

3<0 3<0

1 1 1 1

8 5 8 5

8<5 8<5

5 5 1 2 5 5 1 2

#### Exercise 45

Is there a smallest two digit integer? If so, what is it?

yes,99 yes,99

#### Exercise 46

Is there a smallest two digit real number? If so, what is it?

For the following problems, what integers can replace x x so that the statements are true?

#### Exercise 47

4x7 4x7

##### Solution

4,5,6,or7 4,5,6,or7

3x<1 3x<1

#### Exercise 49

3<x2 3<x2

##### Solution

2,1,0,1,or2 2,1,0,1,or2

#### Exercise 50

The temperature today in Los Angeles was eighty-two degrees. Represent this temperature by real number.

#### Exercise 51

The temperature today in Marbelhead was six degrees below zero. Represent this temperature by real number.

6° 6°

#### Exercise 52

On the number line, how many units between 3 3 and 2?

#### Exercise 53

On the number line, how many units between 4 4 and 0?

4

### Properties of the Real Numbers ((Reference))

#### Exercise 54

a+b=b+a a+b=b+a is an illustration of the



#### Exercise 55

st=ts st=ts is an illustration of the __________ property of __________.

##### Solution

commutative, multiplication

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems.

y+12 y+12

a+4b a+4b

4b+a 4b+a

6x 6x

2( a1 ) 2( a1 )

( a1 )2 ( a1 )2

#### Exercise 60

( 8 )( 4 ) ( 8 )( 4 )

#### Exercise 61

( 6 )( 9 )( 2 ) ( 6 )( 9 )( 2 )

##### Solution

( 9 )( 6 )( 2 )or ( 9 )( 2 )( 6 )or( 6 )( 2 )( 9 )or( 2 )( 9 )( 6 ) ( 9 )( 6 )( 2 )or ( 9 )( 2 )( 6 )or( 6 )( 2 )( 9 )or( 2 )( 9 )( 6 )

#### Exercise 62

( x+y )( xy ) ( x+y )( xy )

#### Exercise 63

##### Solution

Simplify the following problems using the commutative property of multiplication. You need not use the distributive property.

8x3y 8x3y

16ab2c 16ab2c

32abc 32abc

#### Exercise 66

4axyc4d4e 4axyc4d4e

#### Exercise 67

3( x+2 )5( x1 )0( x+6 ) 3( x+2 )5( x1 )0( x+6 )

0

#### Exercise 68

8b( a6 )9a( a4 ) 8b( a6 )9a( a4 )

For the following problems, use the distributive property to expand the expressions.

#### Exercise 69

3( a+4 ) 3( a+4 )

3a+12 3a+12

#### Exercise 70

a( b+3c ) a( b+3c )

#### Exercise 71

2g( 4h+2k ) 2g( 4h+2k )

8gh+4gk 8gh+4gk

#### Exercise 72

( 8m+5n )6p ( 8m+5n )6p

#### Exercise 73

3y( 2x+4z+5w ) 3y( 2x+4z+5w )

##### Solution

6xy+12yz+15wy 6xy+12yz+15wy

#### Exercise 74

( a+2 )( b+2c ) ( a+2 )( b+2c )

#### Exercise 75

( x+y )( 4a+3b ) ( x+y )( 4a+3b )

##### Solution

4ax+3bx+4ay+3by 4ax+3bx+4ay+3by

#### Exercise 76

10 a z ( b z +c ) 10 a z ( b z +c )

### Exponents ((Reference))

For the following problems, write the expressions using exponential notation.

#### Exercise 77

x x to the fifth.

x 5 x 5

#### Exercise 78

( y+2 ) ( y+2 ) cubed.

#### Exercise 79

( a+2b ) ( a+2b ) squared minus ( a+3b ) ( a+3b ) to the fourth.

##### Solution

( a+2b ) 2 ( a+3b ) 4 ( a+2b ) 2 ( a+3b ) 4

#### Exercise 80

x x cubed plus 2 times ( yx ) ( yx ) to the seventh.

aaaaaaa aaaaaaa

a 7 a 7

2222 2222

#### Exercise 83

( 8 )( 8 )( 8 )( 8 )xxxyyyyy ( 8 )( 8 )( 8 )( 8 )xxxyyyyy

##### Solution

( 8 ) 4 x 3 y 5 ( 8 ) 4 x 3 y 5

#### Exercise 84

( x9 )( x9 )+( 3x+1 )( 3x+1 )( 3x+1 ) ( x9 )( x9 )+( 3x+1 )( 3x+1 )( 3x+1 )

#### Exercise 85

2zzyzyyy+7zzyz ( a6 ) 2 ( a6 ) 2zzyzyyy+7zzyz ( a6 ) 2 ( a6 )

##### Solution

2 y 4 z 3 +7y z 3 ( a6 ) 3 2 y 4 z 3 +7y z 3 ( a6 ) 3

For the following problems, expand the terms so that no exponents appear.

x 3 x 3

3 x 3 3 x 3

3xxx 3xxx

7 3 x 2 7 3 x 2

#### Exercise 89

( 4b ) 2 ( 4b ) 2

4b·4b 4b·4b

#### Exercise 90

(6 a 2 ) 3 (5c4) 2 (6 a 2 ) 3 (5c4) 2

#### Exercise 91

( x 3 +7) 2 ( y 2 3) 3 (z+10) ( x 3 +7) 2 ( y 2 3) 3 (z+10)

##### Solution

( xxx+7 )( xxx+7 )( yy3 )( yy3 )( yy3 )( z+10 ) ( xxx+7 )( xxx+7 )( yy3 )( yy3 )( yy3 )( z+10 )

#### Exercise 92

Choose values for a a and b b to show that

1. ( a+b ) 2 ( a+b ) 2 is not always equal to a 2 + b 2 a 2 + b 2 .
2. ( a+b ) 2 ( a+b ) 2 may be equal to a 2 + b 2 a 2 + b 2 .

#### Exercise 93

Choose value for x x to show that

1. ( 4x ) 2 ( 4x ) 2 is not always equal to 4 x 2 4 x 2 .
2. ( 4x ) 2 ( 4x ) 2 may be equal to 4 x 2 4 x 2 .
##### Solution

(a) any value except zero

(b) only zero

### Rules of Exponents ((Reference)) - The Power Rules for Exponents ((Reference))

Simplify the following problems.

4 2 +8 4 2 +8

#### Exercise 95

6 3 +5( 30 ) 6 3 +5( 30 )

366

#### Exercise 96

1 8 + 0 10 + 3 2 ( 4 2 + 2 3 ) 1 8 + 0 10 + 3 2 ( 4 2 + 2 3 )

#### Exercise 97

12 2 +0.3 ( 11 ) 2 12 2 +0.3 ( 11 ) 2

180.3 180.3

#### Exercise 98

3 4 +1 2 2 + 4 2 + 3 2 3 4 +1 2 2 + 4 2 + 3 2

#### Exercise 99

6 2 + 3 2 2 2 +1 + ( 1+4 ) 2 2 3 1 4 2 5 4 2 6 2 + 3 2 2 2 +1 + ( 1+4 ) 2 2 3 1 4 2 5 4 2

10

a 4 a 3 a 4 a 3

#### Exercise 101

2 b 5 2 b 3 2 b 5 2 b 3

4 b 8 4 b 8

#### Exercise 102

4 a 3 b 2 c 8 3a b 2 c 0 4 a 3 b 2 c 8 3a b 2 c 0

#### Exercise 103

(6 x 4 y 10 )(x y 3 ) (6 x 4 y 10 )(x y 3 )

##### Solution

6 x 5 y 13 6 x 5 y 13

#### Exercise 104

(3xy z 2 )(2 x 2 y 3 )(4 x 2 y 2 z 4 ) (3xy z 2 )(2 x 2 y 3 )(4 x 2 y 2 z 4 )

#### Exercise 105

( 3a ) 4 ( 3a ) 4

81 a 4 81 a 4

#### Exercise 106

( 10xy ) 2 ( 10xy ) 2

#### Exercise 107

( x 2 y 4 ) 6 ( x 2 y 4 ) 6

##### Solution

x 12 y 24 x 12 y 24

#### Exercise 108

( a 4 b 7 c 7 z 12 ) 9 ( a 4 b 7 c 7 z 12 ) 9

#### Exercise 109

( 3 4 x 8 y 6 z 0 a 10 b 15 ) 2 ( 3 4 x 8 y 6 z 0 a 10 b 15 ) 2

##### Solution

9 16 x 16 y 12 a 20 b 30 9 16 x 16 y 12 a 20 b 30

x 8 x 5 x 8 x 5

#### Exercise 111

14 a 4 b 6 c 7 2a b 3 c 2 14 a 4 b 6 c 7 2a b 3 c 2

##### Solution

7 a 3 b 3 c 5 7 a 3 b 3 c 5

#### Exercise 112

11 x 4 11 x 4 11 x 4 11 x 4

#### Exercise 113

x 4 x 10 x 3 x 4 x 10 x 3

x 11 x 11

#### Exercise 114

a 3 b 7 a 9 b 6 a 5 b 10 a 3 b 7 a 9 b 6 a 5 b 10

#### Exercise 115

( x 4 y 6 z 10 ) 4 ( x y 5 z 7 ) 3 ( x 4 y 6 z 10 ) 4 ( x y 5 z 7 ) 3

##### Solution

x 13 y 9 z 19 x 13 y 9 z 19

#### Exercise 116

( 2x1 ) 13 ( 2x+5 ) 5 ( 2x1 ) 10 ( 2x+5 ) ( 2x1 ) 13 ( 2x+5 ) 5 ( 2x1 ) 10 ( 2x+5 )

#### Exercise 117

( 3 x 2 4 y 3 ) 2 ( 3 x 2 4 y 3 ) 2

##### Solution

9 x 4 16 y 6 9 x 4 16 y 6

#### Exercise 118

( x+y ) 9 ( xy ) 4 ( x+y ) 3 ( x+y ) 9 ( xy ) 4 ( x+y ) 3

x n x m x n x m

x n+m x n+m

#### Exercise 120

a n+2 a n+4 a n+2 a n+4

#### Exercise 121

6 b 2n+7 8 b 5n+2 6 b 2n+7 8 b 5n+2

##### Solution

48 b 7n+9 48 b 7n+9

#### Exercise 122

18 x 4n+9 2 x 2n+1 18 x 4n+9 2 x 2n+1

#### Exercise 123

( x 5t y 4r ) 7 ( x 5t y 4r ) 7

##### Solution

x 35t y 28r x 35t y 28r

#### Exercise 124

( a 2n b 3m c 4p ) 6r ( a 2n b 3m c 4p ) 6r

u w u k u w u k

u wk u wk

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