Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Derived copy of Fundamentals of Mathematics » Grouping Symbols and the Order of Operations

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • CCQ display tagshide tags

    This module is included in aLens by: Community College of QatarAs a part of collection: "Fundamentals of Mathematics"

    Comments:

    "Used as supplemental materials for developmental math courses."

    Click the "CCQ" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

  • College Open Textbooks display tagshide tags

    This module is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook CollaborativeAs a part of collection: "Fundamentals of Mathematics"

    Comments:

    "Reviewer's Comments: 'I would recommend this text for a basic math course for students moving on to elementary algebra. The information in most chapters is useful, very clear, and easily […]"

    Click the "College Open Textbooks" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Fundamentals of Mathematics"

    Comments:

    "Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • UniqU content

    This module is included inLens: UniqU's lens
    By: UniqU, LLCAs a part of collection: "Fundamentals of Mathematics"

    Click the "UniqU content" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Grouping Symbols and the Order of Operations

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses grouping symbols and the order of operations. By the end of the module students should be able to understand the use of grouping symbols, understand and be able to use the order of operations and use the calculator to determine the value of a numerical expression.

Section Overview

  • Grouping Symbols
  • Multiple Grouping Symbols
  • The Order of Operations
  • Calculators

Grouping Symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:

( ), [ ], { },
   

Parentheses: ( )
Brackets: [ ]
Braces: { }
Bar:

   

In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.

Sample Set A

If possible, determine the value of each of the following.

Example 1

9 + ( 3 8 ) 9 + ( 3 8 ) size 12{9+ \( 3 cdot 8 \) } {}

Since 3 and 8 are within parentheses, they are to be combined first.

9+(38) =9+24 =33 9+(38) =9+24 =33

Thus,

9 + ( 3 8 ) = 33 9 + ( 3 8 ) = 33 size 12{9+ \( 3 cdot 8 \) ="33"} {}

Example 2

( 10 ÷ 0 ) 6 ( 10 ÷ 0 ) 6 size 12{ \( "10"÷0 \) cdot 6} {}

Since 10÷010÷0 size 12{"10" div 0} {} is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."

Practice Set A

If possible, determine the value of each of the following.

Exercise 1

16(32)16(32) size 12{"16"- \( 3 cdot 2 \) } {}

Solution

10

Exercise 2

5+(79)5+(79) size 12{5+ \( 7 cdot 9 \) } {}

Solution

68

Exercise 3

(4+8)2(4+8)2 size 12{ \( 4+8 \) cdot 2} {}

Solution

24

Exercise 4

28÷(1811)28÷(1811) size 12{"28"÷ \( "18"-"11" \) } {}

Solution

4

Exercise 5

(33÷3)11(33÷3)11 size 12{ \( "33"÷3 \) -"11"} {}

Solution

0

Exercise 6

4+(0÷0)4+(0÷0) size 12{4+ \( 0÷0 \) } {}

Solution

not possible (indeterminant)

Multiple Grouping Symbols

When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.

Sample Set B

Determine the value of each of the following.

Example 3

2 + ( 8 3 ) ( 5 + 6 ) 2 + ( 8 3 ) ( 5 + 6 ) size 12{2+ \( 8 cdot 3 \) - \( 5+6 \) } {}

Combine 8 and 3 first, then combine 5 and 6.

2+24-11 Now combine left to right. 26-11 15 2+24-11 Now combine left to right. 26-11 15

Example 4

10 + [ 30 ( 2 9 ) ] 10 + [ 30 ( 2 9 ) ] size 12{"10"+ \[ "30"- \( 2 cdot 9 \) \] } {}

Combine 2 and 9 since they occur in the innermost set of parentheses.

10+[30-18] Now combine 30 and 18. 10+12 22 10+[30-18] Now combine 30 and 18. 10+12 22

Practice Set B

Determine the value of each of the following.

Exercise 7

(17+8)+(9+20)(17+8)+(9+20) size 12{ \( "17"+8 \) + \( 9+"20" \) } {}

Solution

54

Exercise 8

(556)(132)(556)(132) size 12{ \( "55"-6 \) - \( "13" cdot 2 \) } {}

Solution

23

Exercise 9

23+(12÷4)(112)23+(12÷4)(112) size 12{"23"+ \( "12"÷4 \) - \( "11" cdot 2 \) } {}

Solution

4

Exercise 10

86+[14÷(108)]86+[14÷(108)] size 12{"86"+ \[ "14"÷ \( "10"-8 \) \] } {}

Solution

93

Exercise 11

31+{9+[1+(352)]}31+{9+[1+(352)]} size 12{"31"+ lbrace 9+ \[ 1+ \( "35"-2 \) \] rbrace } {}

Solution

74

Exercise 12

{6[24÷(42)]}3{6[24÷(42)]}3 size 12{ lbrace 6- \[ "24"÷ \( 4 cdot 2 \) \] rbrace rSup { size 8{3} } } {}

Solution

27

The Order of Operations

Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of 3+523+52 size 12{3+5 cdot 2} {}. We could do either of two things:

Add 3 and 5, then multiply this sum by 2.

3+52 =82 =16 3+52 =82 =16

Multiply 5 and 2, then add 3 to this product.

3+52 =3+10 =13 3+52 =3+10 =13

We now have two values for one number. To determine the correct value, we must use the accepted order of operations.

Order of Operations

  1. Perform all operations inside grouping symbols, beginning with the innermost set, in the order 2, 3, 4 described below,
  2. Perform all exponential and root operations.
  3. Perform all multiplications and divisions, moving left to right.
  4. Perform all additions and subtractions, moving left to right.

Sample Set C

Determine the value of each of the following.

Example 5

21+312 Multiply first. 21+36 Add. 57 21+312 Multiply first. 21+36 Add. 57

Example 6

(15-8)+5(6+4). Simplify inside parentheses first. 7+510 Multiply. 7+50 Add. 57 (15-8)+5(6+4). Simplify inside parentheses first. 7+510 Multiply. 7+50 Add. 57

Example 7

63-(4+63)+76-4 Simplify first within the parenthesis by multiplying, then adding. 63-(4+18)+76-4 63-22+76-4 Now perform the additions and subtractions, moving left to right. 41+76-4 Add 41 and 76:41+76=117. 117-4 Subtract 4 from 117: 117-4=113. 113 63-(4+63)+76-4 Simplify first within the parenthesis by multiplying, then adding. 63-(4+18)+76-4 63-22+76-4 Now perform the additions and subtractions, moving left to right. 41+76-4 Add 41 and 76:41+76=117. 117-4 Subtract 4 from 117: 117-4=113. 113

Example 8

7642+15 Evaluate the exponential forms, moving left to right. 7616+1 Multiply 7 and 6:76=42 4216+1 Subtract 16 from 42:4216=26 26+1 Add 26 and 1:26+1=27 27 7642+15 Evaluate the exponential forms, moving left to right. 7616+1 Multiply 7 and 6:76=42 4216+1 Subtract 16 from 42:4216=26 26+1 Add 26 and 1:26+1=27 27

Example 9

6(32+22)+42 Evaluate the exponential forms in the parentheses: 32=9 and 22=4 6(9+4)+42 Add the 9 and 4 in the parentheses:9+4=13 6(13)+42 Evaluate the exponential form:42=16 6(13)+16 Multiply 6 and 13:613=78 78+16 Add 78 and 16:78+16=94 94 6(32+22)+42 Evaluate the exponential forms in the parentheses: 32=9 and 22=4 6(9+4)+42 Add the 9 and 4 in the parentheses:9+4=13 6(13)+42 Evaluate the exponential form:42=16 6(13)+16 Multiply 6 and 13:613=78 78+16 Add 78 and 16:78+16=94 94

Example 10

62+2242+622+13+82102195 Recall that the bar is a grouping symbol. The fraction62+2242+622is equivalent to(62+22)÷(42+622) 36 + 4 16 + 6 4 + 1 + 64 100 19 5 36 + 4 16 + 24 + 1 + 64 100 95 40 40 + 65 5 1 + 13 14 62+2242+622+13+82102195 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } + { {1 rSup { size 8{3} } +8 rSup { size 8{2} } } over {"10" rSup { size 8{2} } -"19" cdot 5} } } {} Recall that the bar is a grouping symbol. The fraction62+2242+622 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } } {}is equivalent to(62+22)÷(42+622) size 12{ \( 6 rSup { size 8{2} } +2 rSup { size 8{2} } \) ÷ \( 4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } \) } {} 36 + 4 16 + 6 4 + 1 + 64 100 19 5 size 12{ { {"36"+4} over {"16"+6 cdot 4} } + { {1+"64"} over {"100"-"19" cdot 5} } } {} 36 + 4 16 + 24 + 1 + 64 100 95 size 12{ { {"36"+4} over {"16"+"24"} } + { {1+"64"} over {"100"-"95"} } } {} 40 40 + 65 5 size 12{ { {"40"} over {"40"} } + { {"65"} over {5} } } {} 1 + 13 size 12{1+"13"} {} 14 size 12{"14"} {}

Practice Set C

Determine the value of each of the following.

Exercise 13

8+(327)8+(327) size 12{8+ \( "32"-7 \) } {}

Solution

33

Exercise 14

(34+1823)+11(34+1823)+11 size 12{ \( "34"+"18"-2 cdot 3 \) +"11"} {}

Solution

57

Exercise 15

8(10)+4(2+3)(20+315+405)8(10)+4(2+3)(20+315+405) size 12{8 \( "10" \) +4 \( 2+3 \) - \( "20"+3 cdot "15"+"40"-5 \) } {}

Solution

0

Exercise 16

58+422258+4222 size 12{5 cdot 8+4 rSup { size 8{2} } -2 rSup { size 8{2} } } {}

Solution

52

Exercise 17

4(6233)÷(424)4(6233)÷(424) size 12{4 \( 6 rSup { size 8{2} } -3 rSup { size 8{3} } \) ÷ \( 4 rSup { size 8{2} } -4 \) } {}

Solution

3

Exercise 18

(8+93)÷7+5(8÷4+7+35)(8+93)÷7+5(8÷4+7+35) size 12{ \( 8+9 cdot 3 \) ÷7+5 cdot \( 8÷4+7+3 cdot 5 \) } {}

Solution

125

Exercise 19

33+236229+582+247232÷83+1823333+236229+582+247232÷83+18233 size 12{ { {3 rSup { size 8{3} } +2 rSup { size 8{3} } } over {6 rSup { size 8{2} } -"29"} } +5 left ( { {8 rSup { size 8{2} } +2 rSup { size 8{4} } } over {7 rSup { size 8{2} } -3 rSup { size 8{2} } } } right )÷ { {8 cdot 3+1 rSup { size 8{8} } } over {2 rSup { size 8{3} } -3} } } {}

Solution

7

Calculators

Using a calculator is helpful for simplifying computations that involve large num­bers.

Sample Set D

Use a calculator to determine each value.

Example 11

9, 842 + 56 85 9, 842 + 56 85 size 12{9,"842"+"56" cdot "85"} {}

Table 1
  Key   Display Reads
Perform the multiplication first. Type 56 56
  Press × 56
  Type 85 85
Now perform the addition. Press + 4760
  Type 9842 9842
  Press = 14602

The display now reads 14,602.

Example 12

42 ( 27 + 18 ) + 105 ( 810 ÷ 18 ) 42 ( 27 + 18 ) + 105 ( 810 ÷ 18 ) size 12{"42" \( "27"+"18" \) +"105" \( "810"÷"18" \) } {}

Table 2
  Key   Display Reads
Operate inside the parentheses Type 27 27
  Press + 27
  Type 18 18
  Press = 45
Multiply by 42. Press × 45
  Type 42 42
  Press = 1890

Place this result into memory by pressing the memory key.

Table 3
  Key   Display Reads
Now operate in the other parentheses. Type 810 810
  Press ÷ 810
  Type 18 18
  Press = 45
Now multiply by 105. Press × 45
  Type 105 105
  Press = 4725
We are now ready to add these two quantities together. Press + 4725
Press the memory recall key.     1890
  Press = 6615

Thus, 42(27+18)+105(810÷18)=6,61542(27+18)+105(810÷18)=6,615 size 12{"42" \( "27"+"18" \) +"105" \( "810"÷"18" \) =6,"615"} {}

Example 13

16 4 + 37 3 16 4 + 37 3 size 12{"16" rSup { size 8{4} } +"37" rSup { size 8{3} } } {}

Table 4
Nonscientific Calculators
Key   Display Reads
Type 16 16
Press × 16
Type 16 16
Press × 256
Type 16 16
Press × 4096
Type 16 16
Press = 65536
Press the memory key    
Type 37 37
Press × 37
Type 37 37
Press × 1396
Type 37 37
Press × 50653
Press + 50653
Press memory recall key   65536
Press = 116189
Table 5
Calculators with yxyx Key
Key   Display Reads
Type 16 16
Press y x y x size 12{y rSup { size 8{x} } } {} 16
Type 4 4
Press = 4096
Press + 4096
Type 37 37
Press y x y x size 12{y rSup { size 8{x} } } {} 37
Type 3 3
Press = 116189

Thus, 164+373=116,189164+373=116,189 size 12{"16" rSup { size 8{4} } +"37" rSup { size 8{3} } ="116","189"} {}

We can certainly see that the more powerful calculator simplifies computations.

Example 14

Nonscientific calculators are unable to handle calculations involving very large numbers.

85612 21065 85612 21065 size 12{"85612" cdot "21065"} {}

Table 6
Key   Display Reads
Type 85612 85612
Press × 85612
Type 21065 21065
Press =  

This number is too big for the display of some calculators and we'll probably get some kind of error message. On some scientific calculators such large numbers are coped with by placing them in a form called "scientific notation." Others can do the multiplication directly. (1803416780)

Practice Set D

Use a calculator to find each value.

Exercise 20

9,285+86(49)9,285+86(49) size 12{9,"285"+"86" \( "49" \) } {}

Solution

13,499

Exercise 21

55(8426)+120(512488)55(8426)+120(512488) size 12{"55" \( "84"-"26" \) +"120" \( "512"-"488" \) } {}

Solution

6,070

Exercise 22

10631741063174 size 12{"106" rSup { size 8{3} } -"17" rSup { size 8{4} } } {}

Solution

1,107,495

Exercise 23

6,05336,0533 size 12{6,"053" rSup { size 8{3} } } {}

Solution

This number is too big for a nonscientific calculator. A scientific calculator will probably give you 2.217747109×10112.217747109×1011 size 12{2 "." "217747109"´"10" rSup { size 8{"11"} } } {}

Exercises

For the following problems, find each value. Check each result with a calculator.

Exercise 24

2+3(8)2+3(8) size 12{2+3 cdot \( 8 \) } {}

Solution

26

Exercise 25

18+7(41)18+7(41) size 12{"18"+7 cdot \( 4-1 \) } {}

Exercise 26

3+8(62)+113+8(62)+11 size 12{3+8 cdot \( 6-2 \) +"11"} {}

Solution

46

Exercise 27

15(88)15(88) size 12{1-5 cdot \( 8-8 \) } {}

Exercise 28

3716237162 size 12{"37"-1 cdot 6 rSup { size 8{2} } } {}

Solution

1

Exercise 29

98÷2÷7298÷2÷72 size 12{"98"÷2÷7 rSup { size 8{2} } } {}

Exercise 30

(4224)23(4224)23 size 12{ \( 4 rSup { size 8{2} } -2 cdot 4 \) -2 rSup { size 8{3} } } {}

Solution

0

Exercise 31

9+149+14 size 12{ sqrt {9} +"14"} {}

Exercise 32

100+8142100+8142 size 12{ sqrt {"100"} + sqrt {"81"} -4 rSup { size 8{2} } } {}

Solution

3

Exercise 33

83+82583+825 size 12{ nroot { size 8{3} } {8} +8-2 cdot 5} {}

Exercise 34

1641+521641+52 size 12{ nroot { size 8{4} } {"16"} -1+5 rSup { size 8{2} } } {}

Solution

26

Exercise 35

6122+4[3(10)+11]6122+4[3(10)+11] size 12{"61"-"22"+4 \[ 3 cdot \( "10" \) +"11" \] } {}

Exercise 36

1214[(4)(5)12]+1621214[(4)(5)12]+162 size 12{"121"-4 cdot \[ \( 4 \) cdot \( 5 \) -"12" \] + { {"16"} over {2} } } {}

Solution

97

Exercise 37

(1+16)37+5(12)(1+16)37+5(12) size 12{ { { \( 1+"16" \) -3} over {7} } +5 cdot \( "12" \) } {}

Exercise 38

8(6+20)8+3(6+16)228(6+20)8+3(6+16)22 size 12{ { {8 cdot \( 6+"20" \) } over {8} } + { {3 cdot \( 6+"16" \) } over {"22"} } } {}

Solution

29

Exercise 39

10[8+2(6+7)]10[8+2(6+7)] size 12{"10" cdot \[ 8+2 cdot \( 6+7 \) \] } {}

Exercise 40

21÷7÷321÷7÷3 size 12{"21"÷7÷3} {}

Solution

1

Exercise 41

1023÷523231023÷52323 size 12{"10" rSup { size 8{2} } cdot 3÷5 rSup { size 8{2} } cdot 3-2 cdot 3} {}

Exercise 42

85÷558585÷5585 size 12{"85"÷5 cdot 5-"85"} {}

Solution

0

Exercise 43

5117+7251235117+725123 size 12{ { {"51"} over {"17"} } +7-2 cdot 5 cdot left ( { {"12"} over {3} } right )} {}

Exercise 44

223+23(62)(3+17)+11(6)223+23(62)(3+17)+11(6) size 12{2 rSup { size 8{2} } cdot 3+2 rSup { size 8{3} } cdot \( 6-2 \) - \( 3+"17" \) +"11" \( 6 \) } {}

Solution

90

Exercise 45

2626+20132626+2013 size 12{"26"-2 cdot left lbrace { {6+"20"} over {"13"} } right rbrace } {}

Exercise 46

2{(7+7)+6[4(8+2)]}2{(7+7)+6[4(8+2)]} size 12{2 cdot lbrace \( 7+7 \) +6 cdot \[ 4 cdot \( 8+2 \) \] rbrace } {}

Solution

508

Exercise 47

0+10(0)+15{43+1}0+10(0)+15{43+1} size 12{0+"10" \( 0 \) +"15" cdot lbrace 4 cdot 3+1 rbrace } {}

Exercise 48

18+7+2918+7+29 size 12{"18"+ { {7+2} over {9} } } {}

Solution

19

Exercise 49

(4+7)(83)(4+7)(83) size 12{ \( 4+7 \) cdot \( 8 - 3 \) } {}

Exercise 50

(6+8)(5+24)(6+8)(5+24) size 12{ \( 6+8 \) cdot \( 5+2 - 4 \) } {}

Solution

144

Exercise 51

(213)(61)(7)+4(6+3)(213)(61)(7)+4(6+3) size 12{ \( "21" - 3 \) cdot \( 6 - 1 \) cdot \( 7 \) +4 \( 6+3 \) } {}

Exercise 52

(10+5)(10+5)4(604)(10+5)(10+5)4(604) size 12{ \( "10"+5 \) cdot \( "10"+5 \) - 4 cdot \( "60" - 4 \) } {}

Solution

1

Exercise 53

628+3(5)(2)+84+(1+8)(1+11)628+3(5)(2)+84+(1+8)(1+11) size 12{6 cdot left lbrace 2 cdot 8+3 right rbrace - \( 5 \) cdot \( 2 \) + { {8} over {4} } + \( 1+8 \) cdot \( 1+"11" \) } {}

Exercise 54

25+3(8+1)25+3(8+1) size 12{2 rSup { size 8{5} } +3 cdot \( 8+1 \) } {}

Solution

52

Exercise 55

34+24(1+5)34+24(1+5) size 12{3 rSup { size 8{4} } +2 rSup { size 8{4} } cdot \( 1+5 \) } {}

Exercise 56

16+08+52(2+8)316+08+52(2+8)3 size 12{1 rSup { size 8{6} } +0 rSup { size 8{8} } +5 rSup { size 8{2} } cdot \( 2+8 \) rSup { size 8{3} } } {}

Solution

25,001

Exercise 57

(7)(16)34+22(17+32)(7)(16)34+22(17+32) size 12{ \( 7 \) cdot \( "16" \) - 3 rSup { size 8{4} } +2 rSup { size 8{2} } cdot \( 1 rSup { size 8{7} } +3 rSup { size 8{2} } \) } {}

Exercise 58

2375223752 size 12{ { {2 rSup { size 8{3} } - 7} over {5 rSup { size 8{2} } } } } {}

Solution

125125 size 12{ { {1} over {"25"} } } {}

Exercise 59

1+62+236+11+62+236+1 size 12{ { { left (1+6 right ) rSup { size 8{2} } +2} over {3 cdot 6+1} } } {}

Exercise 60

621233+43+2325621233+43+2325 size 12{ { {6 rSup { size 8{2} } - 1} over {2 rSup { size 8{3} } - 3} } + { {4 rSup { size 8{3} } +2 cdot 3} over {2 cdot 5} } } {}

Solution

14

Exercise 61

58296257+724224558296257+7242245 size 12{ { {5 left (8 rSup { size 8{2} } - 9 cdot 6 right )} over {2 rSup { size 8{5} } - 7} } + { {7 rSup { size 8{2} } - 4 rSup { size 8{2} } } over {2 rSup { size 8{4} } - 5} } } {}

Exercise 62

(2+1)3+23+11062152252552(2+1)3+23+11062152252552 size 12{ { { \( 2+1 \) rSup { size 8{3} } +2 rSup { size 8{3} } +1 rSup { size 8{"10"} } } over {6 rSup { size 8{2} } } } - { {"15" rSup { size 8{2} } - left [2 cdot 5 right ] rSup { size 8{2} } } over {2 rSup { size 8{4} } - 5} } } {}

Solution

0

Exercise 63

63210222+18(23+72)2(19)3363210222+18(23+72)2(19)33 size 12{ { {6 rSup { size 8{3} } - 2 cdot "10" rSup { size 8{2} } } over {2 rSup { size 8{2} } } } + { {"18" \( 2 rSup { size 8{3} } +7 rSup { size 8{2} } \) } over {2 \( "19" \) - 3 rSup { size 8{3} } } } } {}

Exercise 64

26+10262526+102625 size 12{2 cdot left lbrace 6+ left ["10" rSup { size 8{2} } - 6 sqrt {"25"} right ] right rbrace } {}

Solution

152

Exercise 65

1813236+36431813236+3643 size 12{"181" - 3 cdot left (2 sqrt {"36"} +3 nroot { size 8{3} } {"64"} right )} {}

Exercise 66

28112534210+2228112534210+22 size 12{ { {2 cdot left ( sqrt {"81"} - nroot { size 8{3} } {"125"} right )} over {4 rSup { size 8{2} } - "10"+2 rSup { size 8{2} } } } } {}

Solution

4545 size 12{ { {4} over {5} } } {}

Exercises for Review

Exercise 67

((Reference)) The fact that 0 + any whole number = that particular whole number is an example of which property of addition?

Exercise 68

((Reference)) Find the product. 4,271×6304,271×630 size 12{4,"271" times "630"} {}.

Solution

2,690,730

Exercise 69

((Reference)) In the statement 27÷3=927÷3=9 size 12{"27" div 3=9} {}, what name is given to the result 9?

Exercise 70

((Reference)) What number is the multiplicative identity?

Solution

1

Exercise 71

((Reference)) Find the value of 2424 size 12{2 rSup { size 8{4} } } {}.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks