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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

different between demand and supply
Francis Reply
what is the demand and supply of Qd=40,000-6P Qs=14P-28,000 the equilibrium price
Rocky Reply
3400
Ram
what is economics by Adams smith
Diana Reply
economics by Adams Smith's
Francis
to earn wealth more and more
Ram
what are the poor performance of the monopolies
Thandolwethu Reply
it controls only price or production not both a same time
Ram
Difference between demand and supply
Kareem Reply
demand talks about the consumers and supply also talks about producers
Bright
demand talks about the relationship between price and the quantity demanded for a certain goods and supply talks about the relationship between price and quantity supply of a certain goods .
Mutala
demand show the quality demanded at different price n time whereas supply show the quality ready to sell in market by seller at different price n time.
Ram
describe the issue gain from trade will improve economic growth and living standard.
Allison Reply
discribe gain from trade will improve economic growth and living standard
Allison
discribe gain from trade will improve economic growth and living standard
Allison
what is the outcome of exchange rate?
Allison
gain from international trade to a country which import are elastic n export are inelastic
Ram
what is utility
Roberta Reply
the capacity of a good to satisfy our want.
Ram
what is the relationship between the elasticity of demand and total revenue
Atedago Reply
I don't know
Taofeek
negative. but for unit elasticity of demand, the relationship doesn't exist between them.
Ali
what is income elasticity of demand
Esther Reply
income elasticity of demand measure the degree of responsiveness of demand for a commodity to a small change in the income of the consumer
Atedago
monopolistic competition has all except which of the following basic characteristics?
Sharon Reply
what are the Features of money
Abubakarr Reply
how to calculate the elasticity of supply?
Asnaira Reply
PES=Qs2-Qs1/Qs1÷P2-P1/P1
Soko
why is economics important
Gifted Reply
why is it important
Gifted Reply
for what?
Vicky
economics
Gifted
what is demand
Mutala Reply
Demand Is Anything that A Customer are willing and able to buy at different price level.
Basit
what demand schedule
Mutala
demand schedule is a tabular representation of different quantities of commodities that consumers are willing to purchase at a specific price and time while other factors are constant.
MEOW
A full account of the demand, or perhaps we can say, the state of demand for any goods in a given market at a given time should state what the volume (weekly) of sales would be at each of a series of prices. Such an account, taking the form of a tabular statement, is known as a demand schedule.
MEOW
State the type of elasticity of demand
Emmanuel
price elasticity of demand . and it refers to the degree of responsiveness of change in quantity demanded for a commodity to a change in price if the commodity itself
Mutala
Difference between extinct and extici spicies
Amanpreet Reply
In the number 779,844,205 how many ten millions are there?
TELLY Reply
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?
Khizar 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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