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The number 50 is between what two whole numbers?

Since 7 2 = 49 , 49 = 7.

Since 8 2 = 64 , 64 = 8. Thus,

7 < 50 < 8

Thus, 50 is a number between 7 and 8.

A number line with arrows on each end, labeled at zero, seven and eight. Seven is also labeled as square root of forty-nine and eight is labeled as square root of sixty-four. There is a closed circle at square root of fifty and it is labeled as square root of fifty.

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Practice set b

Write the principal and secondary square roots of each number.

100

100 = 10 and 100 = 10

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121

121 = 11 and 121 = 11

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Use a calculator to obtain a decimal approximation for the two square roots of 35. Round to two decimal places.

5.92  and  5.92

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Meaningful expressions

Since we know that the square of any real number is a positive number or zero, we can see that expressions such as 16 do not describe real numbers. There is no real number that can be squared that will produce −16. For x to be a real number, we must have x 0. In our study of algebra, we will assume that all variables and all expressions in radicands represent nonnegative numbers (numbers greater than or equal to zero).

Sample set c

Write the proper restrictions that must be placed on the variable so that each expression represents a real number.

For x 3 to be a real number, we must have

x 3 0 or x 3

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For 2 m + 7 to be a real number, we must have

2 m + 7 0 or 2 m 7 or m 7 2

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Practice set c

Write the proper restrictions that must be placed on the variable so that each expression represents a real number.

Simplifying square roots

When variables occur in the radicand, we can often simplify the expression by removing the radical sign. We can do so by keeping in mind that the radicand is the square of some other expression. We can simplify a radical by seeking an expression whose square is the radicand. The following observations will help us find the square root of a variable quantity.

Since ( x 3 ) 2 = x 3 · 2 = x 6 , x 3 is a square root of x 6 . Also
Six divided by two is equal to three. There is an arrow pointing  towards six that is labeled as

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Since ( x 4 ) 2 = x 4 · 2 = x 8 , x 4 is a square root of x 8 . Also

Eight divided by two is equal to four. There is an arrow pointing  towards eight that is labeled as

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Since ( x 6 ) 2 = x 6 · 2 = x 12 , x 6 is a square root of x 12 . Also

Twelve divided by two is equal to six. There is an arrow pointing  towards twelve that is labeled as

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These examples suggest the following rule:

If a variable has an even exponent, its square root can be found by dividing that exponent by 2.

The examples of Sample Set B illustrate the use of this rule.

Sample set d

Simplify each expression by removing the radical sign. Assume each variable is nonnegative .

a 2 . We seek an expression whose square is  a 2 .  Since ( a ) 2 = a 2 , a 2 = a Notice that  2 ÷ 2 = 1.

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y 8 . We seek an expression whose square is  y 8 .  Since ( y 4 ) 2 = y 8 , y 8 = y 4 Notice that  8 ÷ 2 = 4.

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25 m 2 n 6 . We seek an expression whose square is  25 m 2 n 6 .  Since ( 5 m n 3 ) 2 = 25 m 2 n 6 , 25 m 2 n 6 = 5 m n 3 Notice that  2 ÷ 2 = 1  and  6 ÷ 2 = 3.

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121 a 10 ( b 1 ) 4 . We seek an expression whose square is  121 a 10 ( b 1 ) 4 . Since [ 11 a 5 ( b 1 ) 2 ] 2 = 121 a 10 ( b 1 ) 4 , 121 a 10 ( b 1 ) 4 = 11 a 5 ( b 1 ) 2 Then, 121 a 10 ( b 1 ) 4 = 11 a 5 ( b 1 ) 2 Notice that  10 ÷ 2 = 5  and  4 ÷ 2 = 2.

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Practice set d

Simplify each expression by removing the radical sign. Assume each variable is nonnegative.

100 x 8 y 12 z 2

10 x 4 y 6 z

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36 ( a + 5 ) 4

6 ( a + 5 ) 2

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225 w 4 ( z 2 1 ) 2

15 w 2 ( z 2 1 )

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0.25 y 6 z 14

0.5 y 3 z 7

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x 2 n , where n is a natural number.

x n

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x 4 n , where n is a natural number.

x 2 n

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Exercises

How many square roots does every positive real number have?

two

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The symbol represents which square root of a number?

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The symbol – represents which square root of a number?

secondary

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For the following problems, find the two square roots of the given number.

1 16

1 4 and 1 4

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25 36

5 6  and  5 6

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0.04

0.2 and 0.2

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1.21

1.1 and 1.1

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For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number."

225

not a real number

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1

not a real number

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For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number.

2 x 8

x 4

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For the following problems, simplify each expression by removing the radical sign.

36 x 22 y 44

6 x 11 y 22

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169 w 4 z 6 ( m 1 ) 2

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25 x 12 ( y 1 ) 4

5 x 6 ( y 1 ) 2

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9 m 6 n 4 ( m + n ) 18

3 m 3 n 2 ( m + n ) 9

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( f 2 ) 2 ( g + 6 ) 4

( f 2 ) ( g + 6 ) 4

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( 2 c 3 ) 6 + ( 5 c + 1 ) 2

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64 r 4 s 22

8 r 2 s 11

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121 a 6 ( a 4 ) 8

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[ ( w + 6 ) 2 ]

w + 6

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[ 4 a 2 b 2 ( c 2 + 8 ) 2 ]

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1.21 h 4 k 4

1.1 h 2 k 2

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169 a 2 b 4 c 6 196 x 4 y 6 z 8

13 a b 2 c 3 14 x 2 y 3 z 4

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[ 81 y 4 ( z 1 ) 2 225 x 8 z 4 w 6 ]

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Exercised for review

( [link] ) Find the quotient. x 2 1 4 x 2 1 ÷ x 1 2 x + 1 .

x + 1 2 x 1

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( [link] ) Find the sum. 1 x + 1 + 3 x + 1 + 2 x 2 1 .

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( [link] ) Solve the equation, if possible: 1 x 2 = 3 x 2 x 2 3 x + 1 .

No solution ; x = 2 is excluded.

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( [link] ) Perform the division: 15 x 3 5 x 2 + 10 x 5 x .

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( [link] ) Perform the division: x 3 5 x 2 + 13 x 21 x 3 .

x 2 2 x + 7

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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
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Damian Reply
absolutely yes
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how to know photocatalytic properties of tio2 nanoparticles...what to do now
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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
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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