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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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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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