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7 6 4 2 + 1 5 Evaluate the exponential forms, moving left to right. 7 6 16 + 1 Multiply 7 and 6: 7 6 = 42 42 16 + 1 Subtract 16 from 42: 42 16 = 26 26 + 1 Add 26 and 1: 26 + 1 = 27 27

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6 ( 3 2 + 2 2 ) + 4 2 Evaluate the exponential forms in the parentheses: 3 2 = 9  and  2 2 = 4 6 ( 9 + 4 ) + 4 2 Add the 9 and 4 in the parentheses: 9 + 4 = 13 6 ( 13 ) + 4 2 Evaluate the exponential form: 4 2 = 16 6 ( 13 ) + 16 Multiply 6 and 13: 6 13 = 78 78 + 16 Add 78 and 16: 78 + 16 = 94 94

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6 2 + 2 2 4 2 + 6 2 2 + 1 3 + 8 2 10 2 19 5 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 fraction 6 2 + 2 2 4 2 + 6 2 2 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 ( 6 2 + 2 2 ) ÷ ( 4 2 + 6 2 2 ) 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"} {}

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Practice set c

Determine the value of each of the following.

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

33

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

57

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

0

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

52

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

3

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

125

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3 3 + 2 3 6 2 29 + 5 8 2 + 2 4 7 2 3 2 ÷ 8 3 + 1 8 2 3 3 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} } } {}

7

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Calculators

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

Sample set d

Use a calculator to determine each value.

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

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.

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

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.

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

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

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
Calculators with y x Key
Key Display Reads
Type 16 16
Press y x size 12{y rSup { size 8{x} } } {} 16
Type 4 4
Press = 4096
Press + 4096
Type 37 37
Press y x size 12{y rSup { size 8{x} } } {} 37
Type 3 3
Press = 116189

Thus, 16 4 + 37 3 = 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.

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

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

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)

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Practice set d

Use a calculator to find each value.

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

13,499

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

6,070

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

1,107,495

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

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

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Exercises

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

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

26

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

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

46

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

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

1

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

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

0

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

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

3

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

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

26

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

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

97

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

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

29

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

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21 ÷ 7 ÷ 3 size 12{"21"÷7÷3} {}

1

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

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

0

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

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

90

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

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

508

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

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

19

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

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

144

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

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

1

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6 2 8 + 3 ( 5 ) ( 2 ) + 8 4 + ( 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 \( 1+"11" \) } {}

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

52

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

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1 6 + 0 8 + 5 2 ( 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} } } {}

25,001

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

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

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

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

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6 2 1 2 3 3 + 4 3 + 2 3 2 5 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} } } {}

14

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5 8 2 9 6 2 5 7 + 7 2 4 2 2 4 5 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} } } {}

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( 2 + 1 ) 3 + 2 3 + 1 10 6 2 15 2 2 5 2 5 5 2 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} } } {}

0

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6 3 2 10 2 2 2 + 18 ( 2 3 + 7 2 ) 2 ( 19 ) 3 3 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} } } } } {}

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

152

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

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2 81 125 3 4 2 10 + 2 2 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} } } } } {}

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

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Exercises for review

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

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( [link] ) Find the product. 4, 271 × 630 size 12{4,"271" times "630"} {} .

2,690,730

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

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( [link] ) What number is the multiplicative identity?

1

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( [link] ) Find the value of 2 4 size 12{2 rSup { size 8{4} } } {} .

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Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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7hours 36 min - 4hours 50 min
Tanis Reply

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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