<< Chapter < Page Chapter >> Page >

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"} {}

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

Practice set b

Find the following roots using only a knowledge of multiplication.

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

8

Got questions? Get instant answers now!

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

10

Got questions? Get instant answers now!

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

4

Got questions? Get instant answers now!

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

2

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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 .)

Got questions? Get instant answers now!

Practice set c

Use a calculator to find the following roots.

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

9

Got questions? Get instant answers now!

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

54

Got questions? Get instant answers now!

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

231

Got questions? Get instant answers now!

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

4

Got questions? Get instant answers now!

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} } } {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

6 · 6 ····· 6 85 factors of 6

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

Got questions? Get instant answers now!

2 · 2 ····· 2 112 factors of 2

Got questions? Get instant answers now!

1 · 1 ····· 1 3,008 factors of 1

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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} } } {}

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

9 size 12{ sqrt {9} } {}

Got questions? Get instant answers now!

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

4

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

8

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

12

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

15

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

6

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

20

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

100

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

60

Got questions? Get instant answers now!

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"} } {}

Got questions? Get instant answers now!

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

34

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

4,158

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

24

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

4

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

81

Got questions? Get instant answers now!

Exercises for review

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

Got questions? Get instant answers now!

( [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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Yes; 0

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

discuss how the following factors such as predation risk, competition and habitat structure influence animal's foraging behavior in essay form
Burnet Reply
location of cervical vertebra
KENNEDY Reply
What are acid
Sheriff Reply
define biology infour way
Happiness Reply
What are types of cell
Nansoh Reply
how can I get this book
Gatyin Reply
what is lump
Chineye Reply
what is cell
Maluak Reply
what is biology
Maluak
what's cornea?
Majak Reply
what are cell
Achol
Explain the following terms . (1) Abiotic factors in an ecosystem
Nomai Reply
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc
Qasim
Define the term Abiotic
Marial
what is biology
daniel Reply
what is diffusion
Emmanuel Reply
passive process of transport of low-molecular weight material according to its concentration gradient
AI-Robot
what is production?
Catherine
hello
Marial
Pathogens and diseases
how did the oxygen help a human being
Achol Reply
how did the nutrition help the plants
Achol Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

Ask