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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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