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We can continue this way to see such roots as fourth roots, fifth roots, sixth roots, and so on.

Reading root notation

There is a symbol used to indicate roots of a number. It is called the radical sign n

The radical sign n

The symbol n is called a radical sign and indicates the nth root of a number.

We discuss particular roots using the radical sign as follows:

Square root

number 2 indicates the square root of the number under the radical sign. It is customary to drop the 2 in the radical sign when discussing square roots. The symbol is understood to be the square root radical sign.

49 size 12{ sqrt {"49"} } {} = 7 since 7 7 = 7 2 = 49 size 12{7 cdot 7=7 rSup { size 8{2} } ="49"} {}

Cube root

number 3 size 12{ nroot { size 8{3} } { ital "number"} } {} indicates the cube root of the number under the radical sign.

8 3 = 2 size 12{ nroot { size 8{3} } {8} =2} {} since 2 2 2 = 2 3 = 8 size 12{2 cdot 2 cdot 2=2 rSup { size 8{3} } =8} {}

Fourth root

number 4 size 12{ nroot { size 8{4} } { ital "number"} } {} indicates the fourth root of the number under the radical sign.

81 4 = 3 size 12{ nroot { size 8{4} } {"81"} =3} {} since 3 3 3 3 = 3 4 = 81 size 12{3 cdot 3 cdot 3 cdot 3=3 rSup { size 8{4} } ="81"} {}

In an expression such as 32 5 size 12{ nroot { size 8{5} } {"32"} } {}

Radical sign

is called the radical sign .

Index

5 is called the index . (The index describes the indicated root.)

Radicand

32 is called the radicand .

Radical

32 5 size 12{ nroot { size 8{5} } {"32"} } {} is called a radical (or radical expression).

Sample set b

Find each root.

25 size 12{ sqrt {"25"} } {} To determine the square root of 25, we ask, "What whole number squared equals 25?" From our experience with multiplication, we know this number to be 5. Thus,

25 = 5 size 12{ sqrt {"25"} =5} {}

Check: 5 5 = 5 2 = 25 size 12{5 cdot 5=5 rSup { size 8{2} } ="25"} {}

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32 5 size 12{ nroot { size 8{5} } {"32"} } {} To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32?" This number is 2.

32 5 = 2 size 12{ nroot { size 8{5} } {"32"} =2} {}

Check: 2 2 2 2 2 = 2 5 = 32 size 12{2 cdot 2 cdot 2 cdot 2 cdot 2=2 rSup { size 8{5} } ="32"} {}

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Practice set b

Find the following roots using only a knowledge of multiplication.

64 size 12{ sqrt {"64"} } {}

8

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100 size 12{ sqrt {"100"} } {}

10

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64 3 size 12{ nroot { size 8{3} } {"64"} } {}

4

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64 6 size 12{ nroot { size 8{6} } {"64"} } {}

2

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Calculators

Calculators with the x size 12{ sqrt {x} } {} , y x size 12{y rSup { size 8{x} } } {} , and 1 / x size 12{1/x} {} keys can be used to find or approximate roots.

Sample set c

Use the calculator to find 121 size 12{ sqrt {"121"} } {}

Display Reads
Type 121 121
Press x size 12{ sqrt {x} } {} 11
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Find 2187 7 .

Display Reads
Type 2187 2187
Press y x size 12{y rSup { size 8{x} } } {} 2187
Type 7 7
Press 1 / x size 12{1/x} {} .14285714
Press = 3

2187 7 = 3 (Which means that 3 7 = 2187 .)

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Practice set c

Use a calculator to find the following roots.

729 3 size 12{ nroot { size 8{3} } {"729"} } {}

9

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8503056 4 size 12{ nroot { size 8{4} } {"8503056"} } {}

54

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53361 size 12{ sqrt {"53361"} } {}

231

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16777216 12 size 12{ nroot { size 8{"12"} } {"16777216"} } {}

4

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Exercises

For the following problems, write the expressions using expo­nential notation.

4 4 size 12{4 cdot 4} {}

4 2 size 12{4 rSup { size 8{2} } } {}

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12 12 size 12{"12" cdot "12"} {}

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9 9 9 9 size 12{9 cdot 9 cdot 9 cdot 9} {}

9 4 size 12{9 rSup { size 8{4} } } {}

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10 10 10 10 10 10 size 12{"10" cdot "10" cdot "10" cdot "10" cdot "10" cdot "10"} {}

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826 826 826 size 12{"826" cdot "826" cdot "826"} {}

826 3 size 12{"826" rSup { size 8{3} } } {}

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3, 021 3, 021 3, 021 3, 021 3, 021 size 12{3,"021" cdot 3,"021" cdot 3,"021" cdot 3,"021" cdot 3,"021"} {}

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6 · 6 ····· 6 85 factors of 6

6 85 size 12{6 rSup { size 8{"85"} } } {}

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2 · 2 ····· 2 112 factors of 2

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1 · 1 ····· 1 3,008 factors of 1

1 3008 size 12{1 rSup { size 8{"3008"} } } {}

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For the following problems, expand the terms. (Do not find the actual value.)

5 3 size 12{5 rSup { size 8{3} } } {}

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7 4 size 12{7 rSup { size 8{4} } } {}

7 7 7 7 size 12{7 cdot 7 cdot 7 cdot 7} {}

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15 2 size 12{"15" rSup { size 8{2} } } {}

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117 5 size 12{"117" rSup { size 8{5} } } {}

117 117 117 117 117 size 12{"117" cdot "117" cdot "117" cdot "117" cdot "117"} {}

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61 6 size 12{"61" rSup { size 8{6} } } {}

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30 2 size 12{"30" rSup { size 8{2} } } {}

30 30 size 12{"30" cdot "30"} {}

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For the following problems, determine the value of each of the powers. Use a calculator to check each result.

3 2 size 12{3 rSup { size 8{2} } } {}

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4 2 size 12{4 rSup { size 8{2} } } {}

4 4 = 16 size 12{4 cdot 4="16"} {}

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1 2 size 12{1 rSup { size 8{2} } } {}

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10 2 size 12{"10" rSup { size 8{2} } } {}

10 10 = 100 size 12{"10" cdot "10"="100"} {}

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11 2 size 12{"11" rSup { size 8{2} } } {}

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12 2 size 12{"12" rSup { size 8{2} } } {}

12 12 = 144 size 12{"12" cdot "12"="144"} {}

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13 2 size 12{"13" rSup { size 8{2} } } {}

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15 2 size 12{"15" rSup { size 8{2} } } {}

15 15 = 225 size 12{"15" cdot "15"="225"} {}

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1 4 size 12{1 rSup { size 8{4} } } {}

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3 4 size 12{3 rSup { size 8{4} } } {}

3 3 3 3 = 81 size 12{3 cdot 3 cdot 3 cdot 3 ="81"} {}

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7 3 size 12{7 rSup { size 8{3} } } {}

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10 3 size 12{"10" rSup { size 8{3} } } {}

10 10 10 = 1, 000 size 12{"10" cdot "10" cdot "10"=1,"000"} {}

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100 2 size 12{"100" rSup { size 8{2} } } {}

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8 3 size 12{8 rSup { size 8{3} } } {}

8 8 8 = 512 size 12{8 cdot 8 cdot 8="512"} {}

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5 5 size 12{5 rSup { size 8{5} } } {}

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9 3 size 12{9 rSup { size 8{3} } } {}

9 9 9 = 729 size 12{9 cdot 9 cdot 9="729"} {}

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6 2 size 12{6 rSup { size 8{2} } } {}

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7 1 size 12{7 rSup { size 8{1} } } {}

7 1 = 7 size 12{7 rSup { size 8{1} } =7} {}

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1 28 size 12{1 rSup { size 8{"28"} } } {}

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2 7 size 12{2 rSup { size 8{7} } } {}

2 2 2 2 2 2 2 = 128 size 12{2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2="128"} {}

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0 5 size 12{0 rSup { size 8{5} } } {}

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8 4 size 12{8 rSup { size 8{4} } } {}

8 8 8 8 = 4, 096 size 12{8 cdot 8 cdot 8 cdot 8=4,"096"} {}

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5 8 size 12{5 rSup { size 8{8} } } {}

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6 9 size 12{6 rSup { size 8{9} } } {}

6 6 6 6 6 6 6 6 6 = 10 , 077 , 696 size 12{6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6="10","077","696"} {}

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25 3 size 12{"25" rSup { size 8{3} } } {}

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42 2 size 12{"42" rSup { size 8{2} } } {}

42 42 = 1, 764 size 12{"42" cdot "42"=1,"764"} {}

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31 3 size 12{"31" rSup { size 8{3} } } {}

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15 5 size 12{"15" rSup { size 8{5} } } {}

15 15 15 15 15 = 759 , 375 size 12{"15" cdot "15" cdot "15" cdot "15" cdot "15"="759","375"} {}

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2 20 size 12{2 rSup { size 8{"20"} } } {}

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816 2 size 12{"816" rSup { size 8{2} } } {}

816 816 = 665 , 856 size 12{"816" cdot "816"="665","856"} {}

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For the following problems, find the roots (using your knowledge of multiplication). Use a calculator to check each result.

9 size 12{ sqrt {9} } {}

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16 size 12{ sqrt {"16"} } {}

4

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36 size 12{ sqrt {"36"} } {}

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64 size 12{ sqrt {"64"} } {}

8

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121 size 12{ sqrt {"121"} } {}

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144 size 12{ sqrt {"144"} } {}

12

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169 size 12{ sqrt {"169"} } {}

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225 size 12{ sqrt {"225"} } {}

15

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27 3 size 12{ nroot { size 8{3} } {"27"} } {}

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32 5 size 12{ nroot { size 8{5} } {"32"} } {}

2

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256 4 size 12{ nroot { size 8{4} } {"256"} } {}

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216 3 size 12{ nroot { size 8{3} } {"216"} } {}

6

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1 7 size 12{ nroot { size 8{7} } {1} } {}

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400 size 12{ sqrt {"400"} } {}

20

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900 size 12{ sqrt {"900"} } {}

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10 , 000 size 12{ sqrt {"10","000"} } {}

100

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324 size 12{ sqrt {"324"} } {}

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3, 600 size 12{ sqrt {3,"600"} } {}

60

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For the following problems, use a calculator with the keys x size 12{ sqrt {x} } {} , y x size 12{y rSup { size 8{x} } } {} , and 1 / x size 12{1/x} {} to find each of the values.

676 size 12{ sqrt {"676"} } {}

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1, 156 size 12{ sqrt {1,"156"} } {}

34

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46 , 225 size 12{ sqrt {"46","225"} } {}

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17 , 288 , 964 size 12{ sqrt {"17","288","964"} } {}

4,158

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3, 375 3 size 12{ nroot { size 8{3} } {3,"375"} } {}

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331 , 776 4 size 12{ nroot { size 8{4} } {"331","776"} } {}

24

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5, 764 , 801 8 size 12{ nroot { size 8{8} } {5,"764","801"} } {}

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16 , 777 , 216 12 size 12{ nroot { size 8{"12"} } {"16","777","216"} } {}

4

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16 , 777 , 216 8 size 12{ nroot { size 8{8} } {"16","777","216"} } {}

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9, 765 , 625 10 size 12{ nroot { size 8{"10"} } {9,"765","625"} } {}

5

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160 , 000 4 size 12{ nroot { size 8{4} } {"160","000"} } {}

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531 , 441 3 size 12{ nroot { size 8{3} } {"531","441"} } {}

81

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Exercises for review

( [link] ) Use the numbers 3, 8, and 9 to illustrate the associative property of addition.

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( [link] ) In the multiplication 8 4 = 32 size 12{8 cdot 4="32"} {} , specify the name given to the num­bers 8 and 4.

8 is the multiplier; 4 is the multiplicand

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( [link] ) Does the quotient 15 ÷ 0 exist? If so, what is it?

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( [link] ) Does the quotient 0 ÷ 15 exist? If so, what is it?

Yes; 0

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( [link] ) Use the numbers 4 and 7 to illustrate the commutative property of multiplication.

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Questions & Answers

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taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
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to drain extracellular fluid all over the body.
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Anatomy is the identification and description of the structures of living things
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Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
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Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
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it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
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37 degrees selcius
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the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
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anaemia is the decrease in RBC count hemoglobin count and PVC count
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Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
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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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