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To simplify a square root expression that does not involve a fraction, we can use the following two rules:

Simplifying square roots without fractions

  1. If a factor of the radicand contains a variable with an even exponent, the square root is obtained by dividing the exponent by 2.
  2. If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.

Sample set a

Simplify each square root.

a 4 .     The exponent is even: 4 2 = 2. The exponent on the square root is 2.

a 4 = a 2

a 6 b 10 .    Both exponents are even: 6 2 = 3 and 10 2 = 5. The exponent on the square root of a 6 is 3. The exponent on the square root if b 10 is 5.

a 6 b 10 = a 3 b 5

y 5 .    The exponent is odd: y 5 = y 4 y . Then

y 5 = y 4 y = y 4 y = y 2 y

36 a 7 b 11 c 20 = 6 2 a 6 a b 10 b c 20 a 7 = a 6 a , b 11 = b 10 b = 6 2 a 6 b 10 c 20 · a b by the commutative property of multiplication. = 6 2 a 6 b 10 c 20 a b by the product property of square roots. = 6 a 3 b 5 c 10 a b

49 x 8 y 3 ( a 1 ) 6 = 7 2 x 8 y 2 y ( a 1 ) 6 = 7 2 x 8 y 2 ( a 1 ) 6 y = 7 x 4 y ( a 1 ) 3 y

75 = 25 · 3 = 5 2 · 3 = 5 2 3 = 5 3

Practice set a

Simplify each square root.

m 8

m 4

h 14 k 22

h 7 k 11

81 a 12 b 6 c 38

9 a 6 b 3 c 19

144 x 4 y 80 ( b + 5 ) 16

12 x 2 y 40 ( b + 5 ) 8

w 5

w 2 w

w 7 z 3 k 13

w 3 z k 6 w z k

27 a 3 b 4 c 5 d 6

3 a b 2 c 2 d 3 3 a c

180 m 4 n 15 ( a 12 ) 15

6 m 2 n 7 ( a 12 ) 7 5 n ( a 12 )

Square roots involving fractions

A square root expression is in simplified form if there are

  1. no perfect squares in the radicand,
  2. no fractions in the radicand, or
  3. 3. no square root expressions in the denominator.

The square root expressions 5 a , 4 3 x y 5 , and 11 m 2 n a 4 2 x 2 are in simplified form.

The square root expressions 3 x 8 , 4 a 4 b 3 5 , and 2 y 3 x are not in simplified form.

Simplifying square roots with fractions

To simplify the square root expression x y ,

  1. Write the expression as x y using the rule x y = x y .
  2. Multiply the fraction by 1 in the form of y y .
  3. Simplify the remaining fraction, x y y .

Rationalizing the denominator

The process involved in step 2 is called rationalizing the denominator. This process removes square root expressions from the denominator using the fact that ( y ) ( y ) = y .

Sample set b

Simplify each square root.

9 25 = 9 25 = 3 5

3 5 = 3 5 = 3 5 · 5 5 = 15 5

9 8 = 9 8 = 9 8 · 8 8 = 3 8 8 = 3 4 · 2 8 = 3 4 2 8 = 3 · 2 2 8 = 3 2 4

k 2 m 3 = k 2 m 3 = k m 3 = k m 2 m = k m 2 m = k m m = k m m · m m = k m m m m = k m m · m = k m m 2

x 2 8 x + 16 = ( x 4 ) 2 = x 4

Practice set b

Simplify each square root.

81 25

9 5

2 7

14 7

4 5

2 5 5

10 4

10 2

9 4

3 2

a 3 6

a 6 a 6

y 4 x 3

y 2 x x 2

32 a 5 b 7

4 a 2 2 a b b 4

( x + 9 ) 2

x + 9

x 2 + 14 x + 49

x + 7

Exercises

For the following problems, simplify each of the radical expressions.

8 b 2

2 b 2

20 a 2

24 x 4

2 x 2 6

27 y 6

a 5

a 2 a

m 7

x 11

x 5 x

y 17

36 n 9

6 n 4 n

49 x 13

100 x 5 y 11

10 x 2 y 5 x y

64 a 7 b 3

5 16 m 6 n 7

20 m 3 n 3 n

8 9 a 4 b 11

3 16 x 3

12 x x

8 25 y 3

12 a 4

2 a 2 3

32 m 8

32 x 7

4 x 3 2 x

12 y 13

50 a 3 b 9

5 a b 4 2 a b

48 p 11 q 5

4 18 a 5 b 17

12 a 2 b 8 2 a b

8 108 x 21 y 3

4 75 a 4 b 6

20 a 2 b 3 3

6 72 x 2 y 4 z 10

b 12

b 6

c 18

a 2 b 2 c 2

a b c

4 x 2 y 2 z 2

9 a 2 b 3

3 a b b

16 x 4 y 5

m 6 n 8 p 12 q 20

m 3 n 4 p 6 q 10

r 2

p 2

p

1 4

1 16

1 4

4 25

9 49

3 7

5 8 3

2 32 3

8 6 3

5 6

2 7

14 7

3 10

4 3

2 3 3

2 5

3 10

30 10

16 a 2 5

24 a 5 7

2 a 2 42 a 7

72 x 2 y 3 5

2 a

2 a a

5 b

6 x 3

6 x x 2

12 y 5

49 x 2 y 5 z 9 25 a 3 b 11

7 x y 2 z 4 a b y z 5 a 2 b 6

27 x 6 y 15 3 3 x 3 y 5

( b + 2 ) 4

( b + 2 ) 2

( a 7 ) 8

( x + 2 ) 6

( x + 2 ) 3

( x + 2 ) 2 ( x + 1 ) 2

( a 3 ) 4 ( a 1 ) 2

( a 3 ) 2 ( a 1 )

( b + 7 ) 8 ( b 7 ) 6

a 2 10 a + 25

( a 5 )

b 2 + 6 b + 9

( a 2 2 a + 1 ) 4

( a 1 ) 4

( x 2 + 2 x + 1 ) 12

Exercises for review

( [link] ) Solve the inequality 3 ( a + 2 ) 2 ( 3 a + 4 )

a 2 3

( [link] ) Graph the inequality 6 x 5 ( x + 1 ) 6.
A horizontal line with arrows on both ends.

( [link] ) Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.

positive; negative

( [link] ) Simplify the complex fraction 5 + 1 x 5 1 x .

( [link] ) Simplify 121 x 4 w 6 z 8 by removing the radical sign.

11 x 2 w 3 z 4

Questions & Answers

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
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
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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