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Roots, Radicals, and Square Root Equations: Division of Square Root Expressions

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: 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 x y = x y

Since this is an equation, we may write it as

x y = x y 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

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.

Example 1

3 7 . 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 3 7 = 3 7 = 3 7 · 7 7 = 3 7 7 = 21 7

Example 2

5 3 . 5 3 .

A direct application of the rule produces 5 3 , 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 5 3 = 5 3 · 3 3 = 5 3 3 = 15 3

Example 3

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

Example 4

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

Example 5

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

Example 6

5a b . 5a b .

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

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

Example 7

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

Example 8

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 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

Practice Set A

Simplify the square root expressions.

Exercise 1

26 13 26 13

Solution

2 2

Exercise 2

7 3 7 3

Solution

21 3 21 3

Exercise 3

80 m 5 n 8 5 m 2 n 80 m 5 n 8 5 m 2 n

Solution

4m n 3 mn 4m n 3 mn

Exercise 4

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

Solution

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

Exercise 5

n+4 n-5 n+4 n-5

Solution

n 2 -n-20 n-5 n 2 -n-20 n-5

Exercise 6

a 2 -6a+8 a-2 a 2 -6a+8 a-2

Solution

a-4 a-4

Exercise 7

x 3 n x n x 3 n x n

Solution

x n x n

Exercise 8

a 3m-5 a m-1 a 3m-5 a m-1

Solution

a m-2 a m-2

Conjugates and Rationalizing the Denominator

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

Conjugate

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

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

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

Example 9

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

Example 10

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

Sample Set B

Simplify the following expressions.

Example 11

3 4+ 6 3 4+ 6 .

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

Example 12

2x 3 - 5x . 2x 3 - 5x .

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

2x 3 5x · 3 + 5x 3 + 5x = 2x ( 3 + 5x ) ( 3 ) 2 ( 5x ) 2 = 2x 3 + 2x 5x 35x = 6x + 10 x 2 35x = 6x +x 10 35x 2x 3 5x · 3 + 5x 3 + 5x = 2x ( 3 + 5x ) ( 3 ) 2 ( 5x ) 2 = 2x 3 + 2x 5x 35x = 6x + 10 x 2 35x = 6x +x 10 35x

Practice Set B

Simplify the following expressions.

Exercise 9

5 9+ 7 5 9+ 7

Solution

45-5 7 74 45-5 7 74

Exercise 10

-2 1- 3x -2 1- 3x

Solution

-2-2 3x 1-3x -2-2 3x 1-3x

Exercise 11

8 3x + 2x 8 3x + 2x

Solution

2 6x -4 x x 2 6x -4 x x

Exercise 12

2m m- 3m 2m m- 3m

Solution

2m + 6 m-3 2m + 6 m-3

Exercises

For the following problems, simplify each expressions.

Exercise 13

28 2 28 2

Solution

14 14

Exercise 14

200 10 200 10

Exercise 15

Exercise 16

96 24 96 24

Exercise 17

Exercise 18

336 21 336 21

Exercise 19

Exercise 20

25 9 25 9

Exercise 21

36 35 36 35

Solution

6 35 35 6 35 35

Exercise 22

225 16 225 16

Exercise 23

49 225 49 225

Solution

7 15 7 15

Exercise 24

3 5 3 5

Exercise 25

3 7 3 7

Solution

21 7 21 7

Exercise 26

1 2 1 2

Exercise 27

5 2 5 2

Solution

10 2 10 2

Exercise 28

11 25 11 25

Exercise 29

15 36 15 36

Solution

15 6 15 6

Exercise 30

5 16 5 16

Exercise 31

7 25 7 25

Solution

7 5 7 5

Exercise 32

32 49 32 49

Exercise 33

50 81 50 81

Solution

5 2 9 5 2 9

Exercise 34

125 x 5 5 x 3 125 x 5 5 x 3

Exercise 35

72 m 7 2 m 3 72 m 7 2 m 3

Solution

6 m 2 6 m 2

Exercise 36

162 a 11 2 a 5 162 a 11 2 a 5

Exercise 37

75 y 10 3 y 4 75 y 10 3 y 4

Solution

5 y 3 5 y 3

Exercise 38

48 x 9 3 x 2 48 x 9 3 x 2

Exercise 39

125 a 14 5 a 5 125 a 14 5 a 5

Solution

5 a 4 a 5 a 4 a

Exercise 40

27 a 10 3 a 5 27 a 10 3 a 5

Exercise 41

108 x 21 3 x 4 108 x 21 3 x 4

Solution

6 x 8 x 6 x 8 x

Exercise 42

48 x 6 y 7 3xy 48 x 6 y 7 3xy

Exercise 43

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

Solution

3a b 3 c 3a b 3 c

Exercise 44

66 m 12 n 15 11m n 8 66 m 12 n 15 11m n 8

Exercise 45

30 p 5 q 14 5 q 7 30 p 5 q 14 5 q 7

Solution

p 2 q 3 6pq p 2 q 3 6pq

Exercise 46

b 5 b 5

Exercise 47

5x 2 5x 2

Solution

10x 2 10x 2

Exercise 48

2 a 3 b 14a 2 a 3 b 14a

Exercise 49

3 m 4 n 3 6m n 5 3 m 4 n 3 6m n 5

Solution

m 2m 2n m 2m 2n

Exercise 50

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

Exercise 51

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

Solution

0

Exercise 52

r+1 r-1 r+1 r-1

Exercise 53

s+3 s-3 s+3 s-3

Solution

s 2 9 s3 s 2 9 s3

Exercise 54

a 2 +3a+2 a+1 a 2 +3a+2 a+1

Exercise 55

x 2 -10x+24 x-4 x 2 -10x+24 x-4

Solution

x6 x6

Exercise 56

x 2 -2x-8 x+2 x 2 -2x-8 x+2

Exercise 57

x 2 -4x+3 x-3 x 2 -4x+3 x-3

Solution

x1 x1

Exercise 58

2 x 2 -x-1 x-1 2 x 2 -x-1 x-1

Exercise 59

-5 4+ 5 -5 4+ 5

Solution

20+5 5 11 20+5 5 11

Exercise 60

1 1+ x 1 1+ x

Exercise 61

2 1- a 2 1- a

Solution

2( 1+ a ) 1a 2( 1+ a ) 1a

Exercise 62

6 5 1 6 5 1

Exercise 63

-6 7 +2 -6 7 +2

Solution

2( 7 2 ) 2( 7 2 )

Exercise 64

3 3 - 2 3 3 - 2

Exercise 65

4 6 + 2 4 6 + 2

Solution

6 2 6 2

Exercise 66

5 8 - 6 5 8 - 6

Exercise 67

12 12 - 8 12 12 - 8

Solution

3+ 6 3+ 6

Exercise 68

7x 2- 5x 7x 2- 5x

Exercise 69

6y 1+ 3y 6y 1+ 3y

Solution

6y 3y 2 13y 6y 3y 2 13y

Exercise 70

2 3 - 2 2 3 - 2

Exercise 71

a a + b a a + b

Solution

a ab ab a ab ab

Exercise 72

8 3 b 5 4- 2ab 8 3 b 5 4- 2ab

Exercise 73

7x 5x + x 7x 5x + x

Solution

35 7 4 35 7 4

Exercise 74

3y 2y - y 3y 2y - y

Exercises for Review

Exercise 75

((Reference)) Simplify x 8 y 7 ( x 4 y 8 x 3 y 4 ). x 8 y 7 ( x 4 y 8 x 3 y 4 ).

Solution

x 9 y 11 x 9 y 11

Exercise 76

((Reference)) Solve the compound inequality 875x23. 875x23.

Exercise 77

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

Solution

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

Exercise 78

((Reference)) The symbol x x represents which square root of the number x,x0 x,x0 ?

Exercise 79

((Reference)) Simplify a 2 +8a+16 a 2 +8a+16 .

Solution

a+4 a+4

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