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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: be able to use the division property of square roots, the method of rationalizing the denominator, and conjugates to divide square roots.

Overview

  • The Division Property of Square Roots
  • Rationalizing the Denominator
  • Conjugates and Rationalizing the Denominator

The division property of square roots

In our work with simplifying square root expressions, we noted that

x y = x y

Since this is an equation, we may write it as

x y = x y

To divide two square root expressions, we use the division property of square roots.

The division property x y = x y

x y = x y

The quotient of the square roots is the square root of the quotient.

Rationalizing the denominator

As we can see by observing the right side of the equation governing the division of square roots, the process may produce a fraction in the radicand. This means, of course, that the square root expression is not in simplified form. It is sometimes more useful to rationalize the denominator of a square root expression before actually performing the division.

Sample set a

Simplify the square root expressions.

3 7 .

This radical expression is not in simplified form since there is a fraction under the radical sign. We can eliminate this problem using the division property of square roots.

3 7 = 3 7 = 3 7 · 7 7 = 3 7 7 = 21 7

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5 3 .

A direct application of the rule produces 5 3 , which must be simplified. Let us rationalize the denominator before we perform the division.

5 3 = 5 3 · 3 3 = 5 3 3 = 15 3

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21 7 = 21 7 = 3 .

The rule produces the quotient quickly. We could also rationalize the denominator first and produce the same result.

21 7 = 21 7 · 7 7 = 21 · 7 7 = 3 · 7 · 7 7 = 3 · 7 2 7 = 7 3 7 = 3

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80 x 9 5 x 4 = 80 x 9 5 x 4 = 16 x 5 = 16 x 4 x = 4 x 2 x

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50 a 3 b 7 5 a b 5 = 50 a 3 b 7 5 a b 5 = 10 a 2 b 2 = a b 10

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5 a b .

Some observation shows that a direct division of the radicands will produce a fraction. This suggests that we rationalize the denominator first.

5 a b = 5 a b · b b = 5 a b b = 5 a b b

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m - 6 m + 2 = m - 6 m + 2 · m + 2 m + 2 = m 2 - 4 m - 12 m + 2

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y 2 - y - 12 y + 3 = y 2 - y - 12 y + 3 = ( y + 3 ) ( y - 4 ) ( y + 3 ) = ( y + 3 ) ( y - 4 ) ( y + 3 ) = y - 4

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Practice set a

Simplify the square root expressions.

80 m 5 n 8 5 m 2 n

4 m n 3 m n

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196 ( x + 7 ) 8 2 ( x + 7 ) 3

7 ( x + 7 ) 2 2 ( x + 7 )

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n + 4 n - 5

n 2 - n - 20 n - 5

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a 2 - 6 a + 8 a - 2

a - 4

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a 3 m - 5 a m - 1

a m - 2

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Conjugates and rationalizing the denominator

To perform a division that contains a binomial in the denominator, such as 3 4 + 6 , we multiply the numerator and denominator by a conjugate of the denominator.

Conjugate

A conjugate of the binomial a + b is a - b . Similarly, a conjugate of a - b is a + b .

Notice that when the conjugates a + b and a - b are multiplied together, they produce a difference of two squares.

( a + b ) ( a - b ) = a 2 - a b + a b - b 2 = a 2 - b 2

This principle helps us eliminate square root radicals, as shown in these examples that illustrate finding the product of conjugates.

( 5 + 2 ) ( 5 - 2 ) = 5 2 - ( 2 ) 2 = 25 - 2 = 23

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( 6 - 7 ) ( 6 + 7 ) = ( 6 ) 2 - ( 7 ) 2 = 6 - 7 = - 1

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Sample set b

Simplify the following expressions.

3 4 + 6 .

The conjugate of the denominator is 4 - 6. Multiply the fraction by 1 in the form of 4 - 6 4 - 6 . 3 4 + 6 · 4 - 6 4 - 6 = 3 ( 4 - 6 ) 4 2 - ( 6 ) 2 = 12 - 3 6 16 - 6 = 12 - 3 6 10

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2 x 3 - 5 x .

The conjugate of the denominator is 3 + 5 x . Multiply the fraction by 1 in the form of 3 + 5 x 3 + 5 x .

2 x 3 5 x · 3 + 5 x 3 + 5 x = 2 x ( 3 + 5 x ) ( 3 ) 2 ( 5 x ) 2 = 2 x 3 + 2 x 5 x 3 5 x = 6 x + 10 x 2 3 5 x = 6 x + x 10 3 5 x

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Practice set b

Simplify the following expressions.

- 2 1 - 3 x

- 2 - 2 3 x 1 - 3 x

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8 3 x + 2 x

2 6 x - 4 x x

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2 m m - 3 m

2 m + 6 m - 3

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Exercises

For the following problems, simplify each expressions.

45 a 3 b 8 c 2 5 a b 2 c

3 a b 3 c

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30 p 5 q 14 5 q 7

p 2 q 3 6 p q

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3 m 4 n 3 6 m n 5

m 2 m 2 n

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5 ( p - q ) 6 ( r + s ) 4 25 ( r + s ) 3

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m ( m - 6 ) - m 2 + 6 m 3 m - 7

0

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s + 3 s - 3

s 2 9 s 3

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x 2 - 10 x + 24 x - 4

x 6

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x 2 - 4 x + 3 x - 3

x 1

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- 5 4 + 5

20 + 5 5 11

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2 1 - a

2 ( 1 + a ) 1 a

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- 6 7 + 2

2 ( 7 2 )

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6 y 1 + 3 y

6 y 3 y 2 1 3 y

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a a + b

a a b a b

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Exercises for review

( [link] ) Simplify x 8 y 7 ( x 4 y 8 x 3 y 4 ) .

x 9 y 11

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( [link] ) Solve the compound inequality 8 7 5 x 23.

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( [link] ) Construct the graph of y = 2 3 x - 4.
An xy-plane with gridlines, labeled negative five and five on the both axes.

A graph of a line passing through two points with coordinates three, negative two; and zero, negative five.

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( [link] ) The symbol x represents which square root of the number x , x 0 ?

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( [link] ) Simplify a 2 + 8 a + 16 .

a + 4

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Questions & Answers

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?
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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there is no specific books for beginners but there is book called principle of nanotechnology
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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.
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
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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.
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for screen printed electrodes ?
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of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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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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