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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
7hours 36 min - 4hours 50 min
Tanis 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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