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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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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