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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir 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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