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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

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
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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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