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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

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
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Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
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Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
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BenJay
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Method
I am eliacin, I need your help in maths
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Amoon
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Amoon
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Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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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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