<< 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

price elasticity of demand is the degree of responsiveness of a quantity demanded to the change in price of the commodity in question.
Gladys Reply
what is the importance of learning economics?
Thelma Reply
it helps to make the correct choice
Gladys
it helps firm to produce products that will bring more profit
Gladys
the difference between needs and wants
londiwe Reply
needs are things that we basically can't live without wants are just luxury things
Thelma
needs are things without them we can't live but want are things without we can live
KP
what is education
KP
it's a process in which we give or receiving methodical instructions
Thelma
what is mixed economy
Amex
what is a deadweight loss? how monopoly creates a deadweight loss?
Ashraf Reply
who are u?
Lamine
haha
Cleaford
scarm
nura
what it this
Cleaford
hi y'all
Dope
how does group chat help y'all 🤔
Dope
hi y'all
Dope
how does group chat help y'all 🤔
Dope
how does group chat help y'all 🤔
Dope
to learn from one another
Lamine
oh okay
Dope
😟
Creative
Yes
Lamine
what is type of economic
taiwo Reply
how to understand basics of economics
Aarif Reply
what is demand schedle
Princess Reply
When you make a Scedule of the demand you made
Rodeen
this is helpful for rbi grade b
Prema Reply
What is macroeconomics
Kauna Reply
It's one of the two branches of Economics that deal with the aggregate economy.
Mayen
it's about inflation, occupation, gdp and so on
alberto
What is differences between Microeconomics and Macroeconomic?
Bethrand
microeconomics focuses on the action of individual agents in the economy such as businesses, workers and household. while macroeconomics looks at the economy as a whole. it focuses on broad issues in the economy such as government deficit, economy growth, levels of exports and imports, and
Thelma
inflationary increase in prices
Thelma
a price floor of 24 imposed
Annie Reply
monopolistic competition
bintu Reply
yap
nura
any one there to answer my question
Richard Reply
Fixed Costs per week Variable Costs per bear Rent & Rates of Factory Hire & machines Heating & Lighting Repayment of Bank Loan K100.00 K45.00 K5.00 K50.00 Materials Foam Wages K6.00 K1.00 K1.00 Total K200.00 K8.00
Richard
one of the scarce resources that constrain our behaviour is time. each of us has only 24 hours in a day. how do you go about allocating your time in a given day among completing alternatives? once you choose a most important use of time. why do you not spend all your time to it. use the notion of op
naknak Reply
mohsina mala..Bangla app hobe na
Dipam Reply
mani Baba. First learn the spelling of Economics
Dipam Reply
Economics- The study of how people use their limited resources to tey and satisfy unlimited wants.
Kelly
hmmm
Mani
etar bangla apps hobe na?
Mohsina
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
7hours 36 min - 4hours 50 min
Tanis Reply

Get the best Algebra and trigonometry course in your pocket!





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