<< Chapter < Page Chapter >> Page >
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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set a

Simplify the square root expressions.

80 m 5 n 8 5 m 2 n

4 m n 3 m n

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

n + 4 n - 5

n 2 - n - 20 n - 5

Got questions? Get instant answers now!

a 2 - 6 a + 8 a - 2

a - 4

Got questions? Get instant answers now!

a 3 m - 5 a m - 1

a m - 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set b

Simplify the following expressions.

- 2 1 - 3 x

- 2 - 2 3 x 1 - 3 x

Got questions? Get instant answers now!

8 3 x + 2 x

2 6 x - 4 x x

Got questions? Get instant answers now!

2 m m - 3 m

2 m + 6 m - 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

30 p 5 q 14 5 q 7

p 2 q 3 6 p q

Got questions? Get instant answers now!

3 m 4 n 3 6 m n 5

m 2 m 2 n

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

0

Got questions? Get instant answers now!

s + 3 s - 3

s 2 9 s 3

Got questions? Get instant answers now!

x 2 - 10 x + 24 x - 4

x 6

Got questions? Get instant answers now!

x 2 - 4 x + 3 x - 3

x 1

Got questions? Get instant answers now!

- 5 4 + 5

20 + 5 5 11

Got questions? Get instant answers now!

2 1 - a

2 ( 1 + a ) 1 a

Got questions? Get instant answers now!

- 6 7 + 2

2 ( 7 2 )

Got questions? Get instant answers now!

6 y 1 + 3 y

6 y 3 y 2 1 3 y

Got questions? Get instant answers now!

a a + b

a a b a b

Got questions? Get instant answers now!

Exercises for review

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

x 9 y 11

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( [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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

a + 4

Got questions? Get instant answers now!

Questions & Answers

how do you get the 2/50
Abba Reply
number of sport play by 50 student construct discrete data
Aminu Reply
width of the frangebany leaves on how to write a introduction
Theresa Reply
Solve the mean of variance
Veronica Reply
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.
kenneth
what is error
Yakuba Reply
Is mistake done to something
Vutshila
Hy
anas
hy
What is the life teble
anas
hy
Jibrin
statistics is the analyzing of data
Tajudeen Reply
what is statics?
Zelalem Reply
how do you calculate mean
Gloria Reply
diveving the sum if all values
Shaynaynay
let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?
Fisaye Reply
what is statistics
Akhisani Reply
data collected all over the world
Shaynaynay
construct a less than and more than table
Imad Reply
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Aschalew Reply
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?
Akshay Reply
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits
Akshay
1
Bright
Sorry i want to learn more about this question
Bright
Someone help
Bright
a= 0.20233 b=0.3384
Sufiyan
a
Shaynaynay
How do I interpret level of significance?
Mohd Reply
It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value
Shivam
how skewness and kurtosis are used in statistics
Owen Reply
yes what is it
Taneeya
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask