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Roots, Radicals, and Square Root Equations: Simplifying 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 identify a perfect square, be familiar with the product and quotient properties of square roots, be able to simplify square roots involving and not involving fractions.

Overview

  • Perfect Squares
  • The Product Property of Square Roots
  • The Quotient Property of Square Roots
  • Square Roots Not Involving Fractions
  • Square Roots Involving Fractions

To begin our study of the process of simplifying a square root expression, we must note three facts: one fact concerning perfect squares and two concerning properties of square roots.

Perfect Squares

Perfect Squares

Real numbers that are squares of rational numbers are called perfect squares. The numbers 25 and 1 4 1 4 are examples of perfect squares since 25= 5 2 25= 5 2 and 1 4 = ( 1 2 ) 2 , 1 4 = ( 1 2 ) 2 , and 5 and 1 2 1 2 are rational numbers. The number 2 is not a perfect square since 2= ( 2 ) 2 2= ( 2 ) 2 and 2 2 is not a rational number.

Although we will not make a detailed study of irrational numbers, we will make the following observation:

Any indicated square root whose radicand is not a perfect square is an irrational number.

The numbers 6 , 15 , 6 , 15 , and 3 4 3 4 are each irrational since each radicand ( 6,15, 3 4 ) ( 6,15, 3 4 ) is not a perfect square.

The Product Property of Square Roots

Notice that

9·4 = 36 =6 9·4 = 36 =6      and
9 4 =3·2=6 9 4 =3·2=6

Since both 9·4 9·4 and 9 4 9 4 equal 6, it must be that

9·4 = 9 4 9·4 = 9 4

The Product Property xy = x y xy = x y

This suggests that in general, if x x and y y are positive real numbers,

xy = x y xy = x y

The square root of the product is the product of the square roots.

The Quotient Property of Square Roots

We can suggest a similar rule for quotients. Notice that

36 4 = 9 =3 36 4 = 9 =3      and
36 4 = 6 2 =3 36 4 = 6 2 =3

Since both 36 4 36 4 and 36 4 36 4 equal 3, it must be that

36 4 = 36 4 36 4 = 36 4

The Quotient Property x y = x y x y = x y

This suggests that in general, if x x and y y are positive real numbers,

x y = x y , x y = x y ,       y0 y0

The square root of the quotient is the quotient of the square roots.

CAUTION
It is extremely important to remember that

x+y x + y or xy x y x+y x + y or xy x y

For example, notice that 16+9 = 25 =5, 16+9 = 25 =5, but 16 + 9 =4+3=7. 16 + 9 =4+3=7.

We shall study the process of simplifying a square root expression by distinguishing between two types of square roots: square roots not involving a fraction and square roots involving a fraction.

Square Roots Not Involving Fractions

A square root that does not involve fractions is in simplified form if there are no perfect square in the radicand.

The square roots x, ab , 5mn, 2( a+5 ) x, ab , 5mn, 2( a+5 ) are in simplified form since none of the radicands contains a perfect square.

The square roots x 2 , a 3 = a 2 a x 2 , a 3 = a 2 a are not in simplified form since each radicand contains a perfect square.

To simplify a square root expression that does not involve a fraction, we can use the following two rules:

SIMPLIFYING SQUARE ROOTS WITHOUT FRACTIONS

  1. If a factor of the radicand contains a variable with an even exponent, the square root is obtained by dividing the exponent by 2.
  2. If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.

Sample Set A

Simplify each square root.

Example 1

a 4 . a 4 .     The exponent is even: 4 2 =2. 4 2 =2. The exponent on the square root is 2.

a 4 = a 2 a 4 = a 2

Example 2

a 6 b 10 . a 6 b 10 .    Both exponents are even: 6 2 =3 6 2 =3 and 10 2 =5. 10 2 =5. The exponent on the square root of a 6 a 6 is 3. The exponent on the square root if b 10 b 10 is 5.

a 6 b 10 = a 3 b 5 a 6 b 10 = a 3 b 5

Example 3

y 5 . y 5 .    The exponent is odd: y 5 = y 4 y. y 5 = y 4 y. Then

y 5 = y 4 y = y 4 y = y 2 y y 5 = y 4 y = y 4 y = y 2 y

Example 4

36 a 7 b 11 c 20 = 6 2 a 6 a b 10 b c 20 a 7 = a 6 a, b 11 = b 10 b = 6 2 a 6 b 10 c 20 ·ab by the commutative property of multiplication. = 6 2 a 6 b 10 c 20 ab by the product property of square roots. = 6 a 3 b 5 c 10 ab 36 a 7 b 11 c 20 = 6 2 a 6 a b 10 b c 20 a 7 = a 6 a, b 11 = b 10 b = 6 2 a 6 b 10 c 20 ·ab by the commutative property of multiplication. = 6 2 a 6 b 10 c 20 ab by the product property of square roots. = 6 a 3 b 5 c 10 ab

Example 5

49 x 8 y 3 ( a1 ) 6 = 7 2 x 8 y 2 y ( a1 ) 6 = 7 2 x 8 y 2 ( a1 ) 6 y = 7 x 4 y ( a1 ) 3 y 49 x 8 y 3 ( a1 ) 6 = 7 2 x 8 y 2 y ( a1 ) 6 = 7 2 x 8 y 2 ( a1 ) 6 y = 7 x 4 y ( a1 ) 3 y

Example 6

75 = 25·3 = 5 2 ·3 = 5 2 3 =5 3 75 = 25·3 = 5 2 ·3 = 5 2 3 =5 3

Practice Set A

Simplify each square root.

Exercise 1

m 8 m 8

Solution

m 4 m 4

Exercise 2

h 14 k 22 h 14 k 22

Solution

h 7 k 11 h 7 k 11

Exercise 3

81 a 12 b 6 c 38 81 a 12 b 6 c 38

Solution

9 a 6 b 3 c 19 9 a 6 b 3 c 19

Exercise 4

144 x 4 y 80 ( b+5 ) 16 144 x 4 y 80 ( b+5 ) 16

Solution

12 x 2 y 40 ( b+5 ) 8 12 x 2 y 40 ( b+5 ) 8

Exercise 5

w 5 w 5

Solution

w 2 w w 2 w

Exercise 6

w 7 z 3 k 13 w 7 z 3 k 13

Solution

w 3 z k 6 wzk w 3 z k 6 wzk

Exercise 7

27 a 3 b 4 c 5 d 6 27 a 3 b 4 c 5 d 6

Solution

3a b 2 c 2 d 3 3ac 3a b 2 c 2 d 3 3ac

Exercise 8

180 m 4 n 15 ( a12 ) 15 180 m 4 n 15 ( a12 ) 15

Solution

6 m 2 n 7 ( a12 ) 7 5n( a12 ) 6 m 2 n 7 ( a12 ) 7 5n( a12 )

Square Roots Involving Fractions

A square root expression is in simplified form if there are

  1. no perfect squares in the radicand,
  2. no fractions in the radicand, or
  3. 3. no square root expressions in the denominator.

The square root expressions 5a , 4 3xy 5 , 5a , 4 3xy 5 , and 11 m 2 n a4 2 x 2 11 m 2 n a4 2 x 2 are in simplified form.

The square root expressions 3x 8 , 4 a 4 b 3 5 , 3x 8 , 4 a 4 b 3 5 , and 2y 3x 2y 3x are not in simplified form.

SIMPLIFYING SQUARE ROOTS WITH FRACTIONS

To simplify the square root expression x y , x y ,

  1. Write the expression as x y x y using the rule x y = x y . x y = x y .
  2. Multiply the fraction by 1 in the form of y y . y y .
  3. Simplify the remaining fraction, xy y . xy y .

Rationalizing the Denominator

The process involved in step 2 is called rationalizing the denominator. This process removes square root expressions from the denominator using the fact that ( y )( y )=y. ( y )( y )=y.

Sample Set B

Simplify each square root.

Example 7

9 25 = 9 25 = 3 5 9 25 = 9 25 = 3 5

Example 8

3 5 = 3 5 = 3 5 · 5 5 = 15 5 3 5 = 3 5 = 3 5 · 5 5 = 15 5

Example 9

9 8 = 9 8 = 9 8 · 8 8 = 3 8 8 = 3 4·2 8 = 3 4 2 8 = 3·2 2 8 = 3 2 4 9 8 = 9 8 = 9 8 · 8 8 = 3 8 8 = 3 4·2 8 = 3 4 2 8 = 3·2 2 8 = 3 2 4

Example 10

k 2 m 3 = k 2 m 3 = k m 3 = k m 2 m = k m 2 m = k m m = k m m · m m = k m m m m = k m m·m = k m m 2 k 2 m 3 = k 2 m 3 = k m 3 = k m 2 m = k m 2 m = k m m = k m m · m m = k m m m m = k m m·m = k m m 2

Example 11

x 2 8x+16 = ( x4 ) 2 = x4 x 2 8x+16 = ( x4 ) 2 = x4

Practice Set B

Simplify each square root.

Exercise 9

81 25 81 25

Solution

9 5 9 5

Exercise 10

2 7 2 7

Solution

14 7 14 7

Exercise 11

4 5 4 5

Solution

2 5 5 2 5 5

Exercise 12

10 4 10 4

Solution

10 2 10 2

Exercise 13

9 4 9 4

Solution

3 2 3 2

Exercise 14

a 3 6 a 3 6

Solution

a 6a 6 a 6a 6

Exercise 15

y 4 x 3 y 4 x 3

Solution

y 2 x x 2 y 2 x x 2

Exercise 16

32 a 5 b 7 32 a 5 b 7

Solution

4 a 2 2ab b 4 4 a 2 2ab b 4

Exercise 17

( x+9 ) 2 ( x+9 ) 2

Solution

x+9 x+9

Exercise 18

x 2 +14x+49 x 2 +14x+49

Solution

x+7 x+7

Exercises

For the following problems, simplify each of the radical expressions.

Exercise 19

8 b 2 8 b 2

Solution

2b 2 2b 2

Exercise 20

20 a 2 20 a 2

Exercise 21

24 x 4 24 x 4

Solution

2 x 2 6 2 x 2 6

Exercise 22

27 y 6 27 y 6

Exercise 23

a 5 a 5

Solution

a 2 a a 2 a

Exercise 24

m 7 m 7

Exercise 25

x 11 x 11

Solution

x 5 x x 5 x

Exercise 26

y 17 y 17

Exercise 27

36 n 9 36 n 9

Solution

6 n 4 n 6 n 4 n

Exercise 28

49 x 13 49 x 13

Exercise 29

100 x 5 y 11 100 x 5 y 11

Solution

10 x 2 y 5 xy 10 x 2 y 5 xy

Exercise 30

64 a 7 b 3 64 a 7 b 3

Exercise 31

5 16 m 6 n 7 5 16 m 6 n 7

Solution

20 m 3 n 3 n 20 m 3 n 3 n

Exercise 32

8 9 a 4 b 11 8 9 a 4 b 11

Exercise 33

3 16 x 3 3 16 x 3

Solution

12x x 12x x

Exercise 34

8 25 y 3 8 25 y 3

Exercise 35

12 a 4 12 a 4

Solution

2 a 2 3 2 a 2 3

Exercise 36

32 m 8 32 m 8

Exercise 37

32 x 7 32 x 7

Solution

4 x 3 2x 4 x 3 2x

Exercise 38

12 y 13 12 y 13

Exercise 39

50 a 3 b 9 50 a 3 b 9

Solution

5a b 4 2ab 5a b 4 2ab

Exercise 40

48 p 11 q 5 48 p 11 q 5

Exercise 41

4 18 a 5 b 17 4 18 a 5 b 17

Solution

12 a 2 b 8 2ab 12 a 2 b 8 2ab

Exercise 42

8 108 x 21 y 3 8 108 x 21 y 3

Exercise 43

4 75 a 4 b 6 4 75 a 4 b 6

Solution

20 a 2 b 3 3 20 a 2 b 3 3

Exercise 44

6 72 x 2 y 4 z 10 6 72 x 2 y 4 z 10

Exercise 45

b 12 b 12

Solution

b 6 b 6

Exercise 46

c 18 c 18

Exercise 47

a 2 b 2 c 2 a 2 b 2 c 2

Solution

abc abc

Exercise 48

4 x 2 y 2 z 2 4 x 2 y 2 z 2

Exercise 49

9 a 2 b 3 9 a 2 b 3

Solution

3ab b 3ab b

Exercise 50

16 x 4 y 5 16 x 4 y 5

Exercise 51

m 6 n 8 p 12 q 20 m 6 n 8 p 12 q 20

Solution

m 3 n 4 p 6 q 10 m 3 n 4 p 6 q 10

Exercise 52

r 2 r 2

Exercise 53

Exercise 54

1 4 1 4

Exercise 55

1 16 1 16

Solution

1 4 1 4

Exercise 56

4 25 4 25

Exercise 57

9 49 9 49

Solution

3 7 3 7

Exercise 58

5 8 3 5 8 3

Exercise 59

2 32 3 2 32 3

Solution

8 6 3 8 6 3

Exercise 60

5 6 5 6

Exercise 61

2 7 2 7

Solution

14 7 14 7

Exercise 62

3 10 3 10

Exercise 63

4 3 4 3

Solution

2 3 3 2 3 3

Exercise 64

2 5 2 5

Exercise 65

3 10 3 10

Solution

30 10 30 10

Exercise 66

16 a 2 5 16 a 2 5

Exercise 67

24 a 5 7 24 a 5 7

Solution

2 a 2 42a 7 2 a 2 42a 7

Exercise 68

72 x 2 y 3 5 72 x 2 y 3 5

Exercise 69

2 a 2 a

Solution

2a a 2a a

Exercise 70

5 b 5 b

Exercise 71

6 x 3 6 x 3

Solution

6x x 2 6x x 2

Exercise 72

12 y 5 12 y 5

Exercise 73

49 x 2 y 5 z 9 25 a 3 b 11 49 x 2 y 5 z 9 25 a 3 b 11

Solution

7x y 2 z 4 abyz 5 a 2 b 6 7x y 2 z 4 abyz 5 a 2 b 6

Exercise 74

27 x 6 y 15 3 3 x 3 y 5 27 x 6 y 15 3 3 x 3 y 5

Exercise 75

( b+2 ) 4 ( b+2 ) 4

Solution

( b+2 ) 2 ( b+2 ) 2

Exercise 76

( a7 ) 8 ( a7 ) 8

Exercise 77

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

Solution

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

Exercise 78

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

Exercise 79

( a3 ) 4 ( a1 ) 2 ( a3 ) 4 ( a1 ) 2

Solution

( a3 ) 2 ( a1 ) ( a3 ) 2 ( a1 )

Exercise 80

( b+7 ) 8 ( b7 ) 6 ( b+7 ) 8 ( b7 ) 6

Exercise 81

a 2 10a+25 a 2 10a+25

Solution

( a5 ) ( a5 )

Exercise 82

b 2 +6b+9 b 2 +6b+9

Exercise 83

( a 2 2a+1 ) 4 ( a 2 2a+1 ) 4

Solution

( a1 ) 4 ( a1 ) 4

Exercise 84

( x 2 +2x+1 ) 12 ( x 2 +2x+1 ) 12

Exercises For Review

Exercise 85

((Reference)) Solve the inequality 3( a+2 )2( 3a+4 ) 3( a+2 )2( 3a+4 )

Solution

a 2 3 a 2 3

Exercise 86

((Reference)) Graph the inequality 6x5( x+1 )6. 6x5( x+1 )6.
A horizontal line with arrows on both ends.

Exercise 87

((Reference)) Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.

Solution

positive; negative

Exercise 88

((Reference)) Simplify the complex fraction 5+ 1 x 5 1 x . 5+ 1 x 5 1 x .

Exercise 89

((Reference)) Simplify 121 x 4 w 6 z 8 121 x 4 w 6 z 8 by removing the radical sign.

Solution

11 x 2 w 3 z 4 11 x 2 w 3 z 4

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