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Basic Properties of Real Numbers: Exercise Supplement

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

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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".

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Exercise Supplement

Symbols and Notations ((Reference))

For the following problems, simplify the expressions.

Exercise 1

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

Solution

61

Exercise 2

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

Exercise 3

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

Solution

438

Exercise 4

26÷2−10 26÷2−10

Exercise 5

( 4+17+1 )+4 14−1 ( 4+17+1 )+4 14−1

Solution

2

Exercise 6

51÷3÷7 51÷3÷7

Exercise 7

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

Solution

79

Exercise 8

8( 2⋅12÷13 )+2⋅5⋅11−[ 1+4( 1+2 ) ] 8( 2⋅12÷13 )+2⋅5⋅11−[ 1+4( 1+2 ) ]

Exercise 9

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

Solution

37 48 37 48

Exercise 10

48−3[ 1+17 6 ] 48−3[ 1+17 6 ]

Exercise 11

29+11 6−1 29+11 6−1

Solution

8

Exercise 12

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

Exercise 13

8⋅6 2 + 9⋅9 3 − 10⋅4 5 8⋅6 2 + 9⋅9 3 − 10⋅4 5

Solution

43

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

Exercise 14

22∗6 22∗6

Exercise 15

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

Solution

252>246 252>246

Exercise 16

3( 1.06+2.11 )∗4( 11.01−9.06 ) 3( 1.06+2.11 )∗4( 11.01−9.06 )

Exercise 17

2∗0 2∗0

Solution

2>0 2>0

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

Exercise 18

Strictly less than symbol, and not greater than or equal to symbol.

Exercise 19

> and ≮

Solution

different

Exercise 20

a=b and b=a a=b and b=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.

Exercise 22

8 plus 9

Exercise 23

62 divided by f f

Solution

62 f  or 62÷f 62 f  or 62÷f

Exercise 24

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

Exercise 25

6 times x x , minus 2

Solution

6x−2 6x−2

Exercise 26

x+1 x+1 divided by x−3 x−3

Exercise 27

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

Solution

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

Solution

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.

Exercise 32

3.6 3.6

Exercise 33

−1 3 8 −1 3 8

Solution

A number line with arrows on each end, labeled from negative three to three in increments of one. There is a closed circle at a point between negative two and negative one.

Exercise 34

0

Exercise 35

−4 1 2 −4 1 2

Solution

A number line with arrows on each end, labeled from negative six to one in increments of one. There is a closed circle at a point between negative five and negative four.

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.

Solution

A number line with arrows on each end, labeled from negative ten to ten in increments of two. There are closed circles at negative nine, negative seven, negative five, negative three, negative one, two, four, six, eight, and ten.

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.

Solution

A number line with arrows on each end, labeled from negative ten to ten in increments of two. There is a closed circle at seven and an open circle at negative eight, with a black shaded line connecting the two circles.

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 ( =,<,> ). ( =,<,> ).

Exercise 41

−3 0 −3 0

Solution

−3<0 −3<0

Exercise 42

−1 1 −1 1

Exercise 43

−8 −5 −8 −5

Solution

−8<−5 −8<−5

Exercise 44

−5 −5 1 2 −5 −5 1 2

Exercise 45

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

Solution

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

4≤x≤7 4≤x≤7

Solution

4, 5, 6, or 7 4, 5, 6, or 7

Exercise 48

−3≤x<1 −3≤x<1

Exercise 49

−3<x≤2 −3<x≤2

Solution

−2, −1, 0, 1, or 2 −2, −1, 0, 1, or 2

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.

Solution

−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?

Solution

4

Properties of the Real Numbers ((Reference))

Exercise 54

a+b=b+a a+b=b+a is an illustration of the
property of addition.

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.

Exercise 56

y+12 y+12

Exercise 57

a+4b a+4b

Solution

4b+a 4b+a

Exercise 58

6x 6x

Exercise 59

2( a−1 ) 2( a−1 )

Solution

( a−1 )2 ( a−1 )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 )( x−y ) ( x+y )( x−y )

Exercise 63

△⋅⋄ △⋅⋄

Solution

⋄ ⋅ △ ⋄ ⋅ △

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

Exercise 64

8x3y 8x3y

Exercise 65

16ab2c 16ab2c

Solution

32abc 32abc

Exercise 66

4axyc4d4e 4axyc4d4e

Exercise 67

3( x+2 )5( x−1 )0( x+6 ) 3( x+2 )5( x−1 )0( x+6 )

Solution

0

Exercise 68

8b( a−6 )9a( a−4 ) 8b( a−6 )9a( a−4 )

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

Exercise 69

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

Solution

3a+12 3a+12

Exercise 70

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

Exercise 71

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

Solution

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.

Solution

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 ( y−x ) ( y−x ) to the seventh.

Exercise 81

aaaaaaa aaaaaaa

Solution

a 7 a 7

Exercise 82

2⋅2⋅2⋅2 2⋅2⋅2⋅2

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

( x−9 )( x−9 )+( 3x+1 )( 3x+1 )( 3x+1 ) ( x−9 )( x−9 )+( 3x+1 )( 3x+1 )( 3x+1 )

Exercise 85

2zzyzyyy+7zzyz ( a−6 ) 2 ( a−6 ) 2zzyzyyy+7zzyz ( a−6 ) 2 ( a−6 )

Solution

2 y 4 z 3 +7y z 3 ( a−6 ) 3 2 y 4 z 3 +7y z 3 ( a−6 ) 3

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

Exercise 86

x 3 x 3

Exercise 87

3 x 3 3 x 3

Solution

3xxx 3xxx

Exercise 88

7 3 x 2 7 3 x 2

Exercise 89

( 4b ) 2 ( 4b ) 2

Solution

4b · 4b 4b · 4b

Exercise 90

(6 a 2 ) 3 (5c−4) 2 (6 a 2 ) 3 (5c−4) 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 )( yy−3 )( yy−3 )( yy−3 )( z+10 ) ( xxx+7 )( xxx+7 )( yy−3 )( yy−3 )( yy−3 )( 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.

Exercise 94

4 2 +8 4 2 +8

Exercise 95

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

Solution

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

Solution

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

Solution

10

Exercise 100

a 4 a 3 a 4 a 3

Exercise 101

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

Solution

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

Solution

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

Exercise 110

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

Solution

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

( 2x−1 ) 13 ( 2x+5 ) 5 ( 2x−1 ) 10 ( 2x+5 ) ( 2x−1 ) 13 ( 2x+5 ) 5 ( 2x−1 ) 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 ( x−y ) 4 ( x+y ) 3 ( x+y ) 9 ( x−y ) 4 ( x+y ) 3

Exercise 119

x n ⋅ x m x n ⋅ x m

Solution

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

Exercise 125

u w u k u w u k

Solution

u w−k u w−k

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