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Addition and Subtraction 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: understand the process used in adding and subtracting square roots, be able to add and subtract square roots.

Overview

  • The Logic Behind The Process
  • The Process

The Logic Behind The Process

Now we will study methods of simplifying radical expressions such as

4 3 +8 3 or 5 2x 11 2x +4( 2x +1 ) 4 3 +8 3 or 5 2x 11 2x +4( 2x +1 )

The procedure for adding and subtracting square root expressions will become apparent if we think back to the procedure we used for simplifying polynomial expressions such as

4x+8x or 5a11a+4( a+1 ) 4x+8x or 5a11a+4( a+1 )

The variables x x and a a are letters representing some unknown quantities (perhaps x x represents 3 3 and a a represents 2x 2x ). Combining like terms gives us

4x+8x=12x or 4 3 +8 3 =12 3 and 5a-11a+4( a+1 ) or 5 2x -11 2x +4( 2x +1 ) 5a-11a+4a+4 5 2x -11 2x +4 2x +4 -2a+4 -2 2x +4 4x+8x=12x or 4 3 +8 3 =12 3 and 5a-11a+4( a+1 ) or 5 2x -11 2x +4( 2x +1 ) 5a-11a+4a+4 5 2x -11 2x +4 2x +4 -2a+4 -2 2x +4

The Process

Let’s consider the expression 4 3 +8 3 . 4 3 +8 3 . There are two ways to look at the simplification process:

  1. We are asking, “How many square roots of 3 do we have?”

    4 3 4 3 means we have 4 “square roots of 3”

    8 3 8 3 means we have 8 “square roots of 3”

    Thus, altogether we have 12 “square roots of 3.”
  2. We can also use the idea of combining like terms. If we recall, the process of combining like terms is based on the distributive property

    4x+8x=12x because 4x+8x=( 4+8 )x=12x 4x+8x=12x because 4x+8x=( 4+8 )x=12x

    We could simplify 4 3 +8 3 4 3 +8 3 using the distributive property.

    4 3 +8 3 =( 4+8 ) 3 =12 3 4 3 +8 3 =( 4+8 ) 3 =12 3

Both methods will give us the same result. The first method is probably a bit quicker, but keep in mind, however, that the process works because it is based on one of the basic rules of algebra, the distributive property of real numbers.

Sample Set A

Simplify the following radical expressions.

Example 1

6 10 +11 10 =5 10 6 10 +11 10 =5 10

Example 2

4 32 +5 2 . Simplify 32 . 4 16 · 2 +5 2 = 4 16 2 +5 2 = 4 · 4 2 +5 2 = 16 2 +5 2 = 21 2 4 32 +5 2 . Simplify 32 . 4 16 · 2 +5 2 = 4 16 2 +5 2 = 4 · 4 2 +5 2 = 16 2 +5 2 = 21 2

Example 3

-3x 75 +2x 48 -x 27 . Simple each of the three radicals. = -3x 25 · 3 +2x 16 · 3 -x 9 · 3 = -15x 3 +8x 3 -3x 3 = ( -15x+8x-3x ) 3 = -10x 3 -3x 75 +2x 48 -x 27 . Simple each of the three radicals. = -3x 25 · 3 +2x 16 · 3 -x 9 · 3 = -15x 3 +8x 3 -3x 3 = ( -15x+8x-3x ) 3 = -10x 3

Example 4

5a 24 a 3 -7 54 a 5 + a 2 6a +6a. Simplify each radical. = 5a 4 · 6 · a 2 · a -7 9 · 6 · a 4 · a + a 2 6a +6a = 10 a 2 6a -21 a 2 6a + a 2 6a +6a = ( 10 a 2 -21 a 2 + a 2 ) 6a +6a = -10 a 2 6a +6a Factor out-2a. (This step is optional.) = -2a( 5a 6a -3 ) 5a 24 a 3 -7 54 a 5 + a 2 6a +6a. Simplify each radical. = 5a 4 · 6 · a 2 · a -7 9 · 6 · a 4 · a + a 2 6a +6a = 10 a 2 6a -21 a 2 6a + a 2 6a +6a = ( 10 a 2 -21 a 2 + a 2 ) 6a +6a = -10 a 2 6a +6a Factor out-2a. (This step is optional.) = -2a( 5a 6a -3 )

Practice Set A

Find each sum or difference.

Exercise 1

4 18 5 8 4 18 5 8

Solution

2 2 2 2

Exercise 2

6x 48 +8x 75 6x 48 +8x 75

Solution

64x 3 64x 3

Exercise 3

7 84x 12 189x +2 21x 7 84x 12 189x +2 21x

Solution

48 21x 48 21x

Exercise 4

9 6 8 6 +3 9 6 8 6 +3

Solution

6 +3 6 +3

Exercise 5

a 3 +4a a a 3 +4a a

Solution

5a a 5a a

Exercise 6

4x 54 x 3 + 36 x 2 +3 24 x 5 3x 4x 54 x 3 + 36 x 2 +3 24 x 5 3x

Solution

18 x 2 6x +3x 18 x 2 6x +3x

Sample Set B

Example 5

Finding the product of the square root of seven and the binomial the square root of eight minus three, using the rule for multiplying square root expressions. See the longdesc for a full description.

Example 6

Finding the product of the binomial the square root of two plus the square root of three and the binomial the square root of five plus the square root of twelve, using the rule for multiplying square root expressions. See the longdesc for a full description.

Example 7

Finding the product of the binomial four times the square root of two minus three times the square root of six and the binomial five times the square root of two plus the square root of six, using the rule for multiplying square root expressions. See the longdesc for a full description.

Example 8

3+ 8 3 8 . We'll rationalize the denominator by multiplying this fraction by 1 in the form 3+ 8 3+ 8 . 3+ 8 3 8 · 3+ 8 3+ 8 = (3+ 8 )(3+ 8 ) 3 2 ( 8 ) 2 = 9+3 8 +3 8 + 8 8 98 = 9+6 8 +8 1 = 17+6 8 = 17+6 4·2 = 17+12 2 3+ 8 3 8 . We'll rationalize the denominator by multiplying this fraction by 1 in the form 3+ 8 3+ 8 . 3+ 8 3 8 · 3+ 8 3+ 8 = (3+ 8 )(3+ 8 ) 3 2 ( 8 ) 2 = 9+3 8 +3 8 + 8 8 98 = 9+6 8 +8 1 = 17+6 8 = 17+6 4·2 = 17+12 2

Example 9

2+ 7 4- 3 . Rationalize the denominator by multiplying this fraction by 1 in the from 4+ 3 4+ 3 . 2+ 7 4- 3 · 4+ 3 4+ 3 = ( 2+ 7 )( 4+ 3 ) 4 2 - ( 3 ) 2 = 8+2 3 +4 7 + 21 16-3 = 8+2 3 +4 7 + 21 13 2+ 7 4- 3 . Rationalize the denominator by multiplying this fraction by 1 in the from 4+ 3 4+ 3 . 2+ 7 4- 3 · 4+ 3 4+ 3 = ( 2+ 7 )( 4+ 3 ) 4 2 - ( 3 ) 2 = 8+2 3 +4 7 + 21 16-3 = 8+2 3 +4 7 + 21 13

Practice Set B

Simplify each by performing the indicated operation.

Exercise 7

5 ( 6 4 ) 5 ( 6 4 )

Solution

30 4 5 30 4 5

Exercise 8

( 5 + 7 )( 2 + 8 ) ( 5 + 7 )( 2 + 8 )

Solution

3 10 +3 14 3 10 +3 14

Exercise 9

( 3 2 2 3 )( 4 3 + 8 ) ( 3 2 2 3 )( 4 3 + 8 )

Solution

8 6 12 8 6 12

Exercise 10

4+ 5 3 8 4+ 5 3 8

Solution

12+8 2 +3 5 +2 10 12+8 2 +3 5 +2 10

Exercises

For the following problems, simplify each expression by performing the indicated operation.

Exercise 11

4 5 2 5 4 5 2 5

Solution

2 5 2 5

Exercise 12

10 2 +8 2 10 2 +8 2

Exercise 13

3 6 12 6 3 6 12 6

Solution

15 6 15 6

Exercise 14

10 2 10 10 2 10

Exercise 15

3 7x +2 7x 3 7x +2 7x

Solution

5 7x 5 7x

Exercise 16

6 3a + 3a 6 3a + 3a

Exercise 17

2 18 +5 32 2 18 +5 32

Solution

26 2 26 2

Exercise 18

4 27 3 48 4 27 3 48

Exercise 19

200 128 200 128

Solution

2 2 2 2

Exercise 20

4 300 +2 500 4 300 +2 500

Exercise 21

6 40 +8 80 6 40 +8 80

Solution

12 10 +32 5 12 10 +32 5

Exercise 22

2 120 5 30 2 120 5 30

Exercise 23

8 60 3 15 8 60 3 15

Solution

13 15 13 15

Exercise 24

a 3 3a a a 3 3a a

Exercise 25

4 x 3 +x x 4 x 3 +x x

Solution

3x x 3x x

Exercise 26

2b a 3 b 5 +6a a b 7 2b a 3 b 5 +6a a b 7

Exercise 27

5xy 2x y 3 3 y 2 2 x 3 y 5xy 2x y 3 3 y 2 2 x 3 y

Solution

2x y 2 2xy 2x y 2 2xy

Exercise 28

5 20 +3 45 3 40 5 20 +3 45 3 40

Exercise 29

24 2 54 4 12 24 2 54 4 12

Solution

4 6 8 3 4 6 8 3

Exercise 30

6 18 +5 32 +4 50 6 18 +5 32 +4 50

Exercise 31

8 20 9 125 +10 180 8 20 9 125 +10 180

Solution

5 5

Exercise 32

2 27 +4 3 6 12 2 27 +4 3 6 12

Exercise 33

14 +2 56 3 136 14 +2 56 3 136

Solution

5 14 6 34 5 14 6 34

Exercise 34

3 2 +2 63 +5 7 3 2 +2 63 +5 7

Exercise 35

4ax 3x +2 3 a 2 x 3 +7 3 a 2 x 3 4ax 3x +2 3 a 2 x 3 +7 3 a 2 x 3

Solution

13ax 3x 13ax 3x

Exercise 36

3by 5y +4 5 b 2 y 3 2 5 b 2 y 3 3by 5y +4 5 b 2 y 3 2 5 b 2 y 3

Exercise 37

2 ( 3 +1 ) 2 ( 3 +1 )

Solution

6 + 2 6 + 2

Exercise 38

3 ( 5 3 ) 3 ( 5 3 )

Exercise 39

5 ( 3 2 ) 5 ( 3 2 )

Solution

15 10 15 10

Exercise 40

7 ( 6 3 ) 7 ( 6 3 )

Exercise 41

8 ( 3 + 2 ) 8 ( 3 + 2 )

Solution

2( 6 +2 ) 2( 6 +2 )

Exercise 42

10 ( 10 5 ) 10 ( 10 5 )

Exercise 43

( 1+ 3 )( 2 3 ) ( 1+ 3 )( 2 3 )

Solution

1+ 3 1+ 3

Exercise 44

( 5+ 6 )( 4 6 ) ( 5+ 6 )( 4 6 )

Exercise 45

( 3 2 )( 4 2 ) ( 3 2 )( 4 2 )

Solution

7( 2 2 ) 7( 2 2 )

Exercise 46

( 5+ 7 )( 4 7 ) ( 5+ 7 )( 4 7 )

Exercise 47

( 2 + 5 )( 2 +3 5 ) ( 2 + 5 )( 2 +3 5 )

Solution

17+4 10 17+4 10

Exercise 48

( 2 6 3 )( 3 6 +2 3 ) ( 2 6 3 )( 3 6 +2 3 )

Exercise 49

( 4 5 2 3 )( 3 5 + 3 ) ( 4 5 2 3 )( 3 5 + 3 )

Solution

542 15 542 15

Exercise 50

( 3 8 2 2 )( 4 2 5 8 ) ( 3 8 2 2 )( 4 2 5 8 )

Exercise 51

( 12 +5 3 )( 2 3 2 12 ) ( 12 +5 3 )( 2 3 2 12 )

Solution

42 42

Exercise 52

( 1+ 3 ) 2 ( 1+ 3 ) 2

Exercise 53

( 3+ 5 ) 2 ( 3+ 5 ) 2

Solution

14+6 5 14+6 5

Exercise 54

( 2 6 ) 2 ( 2 6 ) 2

Exercise 55

( 2 7 ) 2 ( 2 7 ) 2

Solution

114 7 114 7

Exercise 56

( 1+ 3x ) 2 ( 1+ 3x ) 2

Exercise 57

( 2+ 5x ) 2 ( 2+ 5x ) 2

Solution

4+4 5x +5x 4+4 5x +5x

Exercise 58

( 3 3x ) 2 ( 3 3x ) 2

Exercise 59

( 8 6b ) 2 ( 8 6b ) 2

Solution

6416 6b +6b 6416 6b +6b

Exercise 60

( 2a+ 5a ) 2 ( 2a+ 5a ) 2

Exercise 61

( 3y 7y ) 2 ( 3y 7y ) 2

Solution

9 y 2 6y 7y +7y 9 y 2 6y 7y +7y

Exercise 62

( 3+ 3 )( 3 3 ) ( 3+ 3 )( 3 3 )

Exercise 63

( 2+ 5 )( 2 5 ) ( 2+ 5 )( 2 5 )

Solution

1 1

Exercise 64

( 8+ 10 )( 8 10 ) ( 8+ 10 )( 8 10 )

Exercise 65

( 6+ 7 )( 6 7 ) ( 6+ 7 )( 6 7 )

Solution

29

Exercise 66

( 2 + 3 )( 2 3 ) ( 2 + 3 )( 2 3 )

Exercise 67

( 5 + 2 )( 5 2 ) ( 5 + 2 )( 5 2 )

Solution

3

Exercise 68

( a + b )( a b ) ( a + b )( a b )

Exercise 69

( x + y )( x y ) ( x + y )( x y )

Solution

xy xy

Exercise 70

2 5+ 3 2 5+ 3

Exercise 71

4 6+ 2 4 6+ 2

Solution

2( 6 2 ) 17 2( 6 2 ) 17

Exercise 72

1 3 2 1 3 2

Exercise 73

1 4 3 1 4 3

Solution

4+ 3 13 4+ 3 13

Exercise 74

8 2 6 8 2 6

Exercise 75

2 3 7 2 3 7

Solution

3+ 7 3+ 7

Exercise 76

5 3+ 3 5 3+ 3

Exercise 77

3 6+ 6 3 6+ 6

Solution

2 3 2 10 2 3 2 10

Exercise 78

2 8 2+ 8 2 8 2+ 8

Exercise 79

4+ 5 4 5 4+ 5 4 5

Solution

21+8 5 11 21+8 5 11

Exercise 80

1+ 6 1 6 1+ 6 1 6

Exercise 81

8 3 2+ 18 8 3 2+ 18

Solution

16+2 3 +24 2 3 6 14 16+2 3 +24 2 3 6 14

Exercise 82

6 2 4+ 12 6 2 4+ 12

Exercise 83

3 2 3 + 2 3 2 3 + 2

Solution

52 6 52 6

Exercise 84

6a 8a 8a + 6a 6a 8a 8a + 6a

Exercise 85

2b 3b 3b + 2b 2b 3b 3b + 2b

Solution

2 6 5 2 6 5

Exercises For Review

Exercise 86

((Reference)) Simplify ( x 5 y 3 x 2 y ) 5 . ( x 5 y 3 x 2 y ) 5 .

Exercise 87

((Reference)) Simplify ( 8 x 3 y ) 2 ( x 2 y 3 ) 4 . ( 8 x 3 y ) 2 ( x 2 y 3 ) 4 .

Solution

64 x 14 y 14 64 x 14 y 14

Exercise 88

((Reference)) Write ( x1 ) 4 ( x1 ) 7 ( x1 ) 4 ( x1 ) 7 so that only positive exponents appear.

Exercise 89

((Reference)) Simplify 27 x 5 y 10 z 3 . 27 x 5 y 10 z 3 .

Solution

3 x 2 y 5 z 3xz 3 x 2 y 5 z 3xz

Exercise 90

((Reference)) Simplify 1 2+ x 1 2+ x by rationalizing the denominator.

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