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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

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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

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

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

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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

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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

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75 = 25 · 3 = 5 2 · 3 = 5 2 3 = 5 3

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Practice set a

Simplify each square root.

81 a 12 b 6 c 38

9 a 6 b 3 c 19

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144 x 4 y 80 ( b + 5 ) 16

12 x 2 y 40 ( b + 5 ) 8

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w 7 z 3 k 13

w 3 z k 6 w z k

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27 a 3 b 4 c 5 d 6

3 a b 2 c 2 d 3 3 a c

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180 m 4 n 15 ( a 12 ) 15

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

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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.

3 5 = 3 5 = 3 5 · 5 5 = 15 5

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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

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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

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

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Practice set b

Simplify each square root.

32 a 5 b 7

4 a 2 2 a b b 4

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Exercises

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

100 x 5 y 11

10 x 2 y 5 x y

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5 16 m 6 n 7

20 m 3 n 3 n

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50 a 3 b 9

5 a b 4 2 a b

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4 18 a 5 b 17

12 a 2 b 8 2 a b

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4 75 a 4 b 6

20 a 2 b 3 3

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9 a 2 b 3

3 a b b

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m 6 n 8 p 12 q 20

m 3 n 4 p 6 q 10

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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

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( a 3 ) 4 ( a 1 ) 2

( a 3 ) 2 ( a 1 )

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( b + 7 ) 8 ( b 7 ) 6

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a 2 10 a + 25

( a 5 )

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( a 2 2 a + 1 ) 4

( a 1 ) 4

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Exercises for review

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

a 2 3

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

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( [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

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

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

11 x 2 w 3 z 4

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Questions & Answers

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s. Reply
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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
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.
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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
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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I'm interested in nanotube
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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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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