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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 concept of square root, be able to distinguish between the principal and secondary square roots of a number, be able to relate square roots and meaningful expressions and to simplify a square root expression.

Overview

  • Square Roots
  • Principal and Secondary Square Roots
  • Meaningful Expressions
  • Simplifying Square Roots

Square Roots

When we studied exponents in Section (Reference), we noted that 4 2 =16 4 2 =16 and ( −4 ) 2 =16. ( −4 ) 2 =16. We can see that 16 is the square of both 4 and −4 −4 . Since 16 comes from squaring 4 or −4 −4 , 4 and −4 −4 are called the square roots of 16. Thus 16 has two square roots, 4 and −4 −4 . Notice that these two square roots are opposites of each other.

We can say that

Square Root

The square root of a positive number x x is a number such that when it is squared the number x x results.

Every positive number has two square roots, one positive square root and one negative square root. Furthermore, the two square roots of a positive number are opposites of each other. The square root of 0 is 0.

Sample Set A

Example 1

The two square roots of 49 are 7 and −7 since

7 2 =49 and ( −7 ) 2 =49 7 2 =49 and ( −7 ) 2 =49

Example 2

The two square roots of 49 64 49 64 are 7 8 7 8 and −7 8 −7 8 since

( 7 8 ) 2 = 7 8  ·  7 8 = 49 64 and ( −7 8 ) 2 = −7 8  ·  −7 8 = 49 64 ( 7 8 ) 2 = 7 8  ·  7 8 = 49 64 and ( −7 8 ) 2 = −7 8  ·  −7 8 = 49 64

Practice Set A

Name both square roots of each of the following numbers.

Exercise 1

36

Solution

6 and −6

Exercise 2

25

Solution

5 and −5

Exercise 3

100

Solution

10 and −10

Exercise 4

64

Solution

8 and −8

Exercise 5

1

Solution

1 and −1

Exercise 6

1 4 1 4

Solution

1 2  and − 1 2 1 2  and − 1 2

Exercise 7

9 16 9 16

Solution

3 4  and − 3 4 3 4  and − 3 4

Exercise 8

0.1 0.1

Solution

0.1 0.1 and −0.1 −0.1

Exercise 9

0.09 0.09

Solution

0.03 0.03 and −0.03 −0.03

Principal and Secondary Square Roots

There is a notation for distinguishing the positive square root of a number x x from the negative square root of x x .

Principal Square Root: x x

If x x is a positive real number, then

x x represents the positive square root of x x . The positive square root of a number is called the principal square root of the number.

Secondary Square Root: − x − x

− x − x represents the negative square root of x x . The negative square root of a number is called the secondary square root of the number.

− x − x indicates the secondary square root of x x .

Radical Sign, Radicand, and Radical

In the expression x, x,

is called a radical sign.

x x is called the radicand.

x x is called a radical.

The horizontal bar that appears attached to the radical sign, , is a grouping symbol that specifies the radicand.

Because x x and − x − x are the two square root of x x ,

( x )( x )=x and ( − x )( − x )=x ( x )( x )=x and ( − x )( − x )=x

Sample Set B

Write the principal and secondary square roots of each number.

Example 3

9. Principal square root is  9 =3. Secondary square root is − 9 =−3. 9. Principal square root is  9 =3. Secondary square root is − 9 =−3.

Example 4

15. Principal square root is  15 . Secondary square root is − 15 . 15. Principal square root is  15 . Secondary square root is − 15 .

Example 5

Use a calculator to obtain a decimal approximation for the two square roots of 34. Round to two decimal places.
On the Calculater Type 34 Press x Display reads: 5.8309519 Round to 5.83. On the Calculater Type 34 Press x Display reads: 5.8309519 Round to 5.83.
Notice that the square root symbol on the calculator is . This means, of course, that a calculator will produce only the positive square root. We must supply the negative square root ourselves.

34 ≈5.83 and − 34 ≈−5.83 34 ≈5.83 and − 34 ≈−5.83
Note: The symbol ≈ means "approximately equal to."

Example 6

The number 50 50 is between what two whole numbers?

Since 7 2 =49,  49 =7. 7 2 =49,  49 =7.

Since 8 2 =64,  64 =8. 8 2 =64,  64 =8. Thus,

 7< 50 <8 7< 50 <8

Thus, 50 50 is a number between 7 and 8.

A number line with arrows on each end, labeled at zero, seven and eight. Seven is also labeled as square root of forty-nine and eight is labeled as square root of sixty-four. There is a closed circle at square root of fifty and it is labeled as square root of fifty.

Practice Set B

Write the principal and secondary square roots of each number.

Exercise 10

100

Solution

100 =10 100 =10 and − 100 =−10 − 100 =−10

Exercise 11

121

Solution

121 =11 121 =11 and − 121 =−11 − 121 =−11

Exercise 12

35

Solution

35 35 and − 35 − 35

Exercise 13

Use a calculator to obtain a decimal approximation for the two square roots of 35. Round to two decimal places.

Solution

5.92 and −5.92 5.92 and −5.92

Meaningful Expressions

Since we know that the square of any real number is a positive number or zero, we can see that expressions such as −16 −16 do not describe real numbers. There is no real number that can be squared that will produce −16. For x x to be a real number, we must have x≥0. x≥0. In our study of algebra, we will assume that all variables and all expressions in radicands represent nonnegative numbers (numbers greater than or equal to zero).

Sample Set C

Write the proper restrictions that must be placed on the variable so that each expression represents a real number.

Example 7

For x−3 x−3 to be a real number, we must have

x−3≥0 or x≥3 x−3≥0 or x≥3

Example 8

For 2m+7 2m+7 to be a real number, we must have

2m+7≥0 or 2m≥−7 or m≥ −7 2 2m+7≥0 or 2m≥−7 or m≥ −7 2

Practice Set C

Write the proper restrictions that must be placed on the variable so that each expression represents a real number.

Exercise 14

x+5 x+5

Solution

x≥−5 x≥−5

Exercise 15

y−8 y−8

Solution

y≥8 y≥8

Exercise 16

3a+2 3a+2

Solution

a≥− 2 3 a≥− 2 3

Exercise 17

5m−6 5m−6

Solution

m≥ 6 5 m≥ 6 5

Simplifying Square Roots

When variables occur in the radicand, we can often simplify the expression by removing the radical sign. We can do so by keeping in mind that the radicand is the square of some other expression. We can simplify a radical by seeking an expression whose square is the radicand. The following observations will help us find the square root of a variable quantity.

Example 9

Since ( x 3 ) 2 = x 3  ·   2 = x 6 , x 3 ( x 3 ) 2 = x 3  ·   2 = x 6 , x 3 is a square root of x 6 . x 6 . Also
 Six divided by two is equal to three. There is an arrow pointing towards six that is labeled as "exponent of x to the sixth power." There is another arrows pointing towards three that is labeled as "exponent of square root of the expression x to the sixth power."

Example 10

Since ( x 4 ) 2 = x 4·2 = x 8 , x 4 ( x 4 ) 2 = x 4·2 = x 8 , x 4 is a square root of x 8 . x 8 . Also
Eight divided by two is equal to four. There is an arrow pointing towards eight that is labeled as "exponent of x to the eighth power." There is another arrows pointing towards four that is labeled as "exponent of square root of the expression x to the eighth power."

Example 11

Since ( x 6 ) 2 = x 6·2 = x 12 , x 6 ( x 6 ) 2 = x 6·2 = x 12 , x 6 is a square root of x 12 . x 12 . Also
Twelve divided by two is equal to six. There is an arrow pointing towards twelve that is labeled as "exponent of x to the twelfth power." There is another arrows pointing towards six that is labeled as "exponent of square root of the expression x to the twelfth power."

These examples suggest the following rule:

If a variable has an even exponent, its square root can be found by dividing that exponent by 2.

The examples of Sample Set B illustrate the use of this rule.

Sample Set D

Simplify each expression by removing the radical sign. Assume each variable is nonnegative.

Example 12

a 2 . We seek an expression whose square is  a 2 . Since   ( a ) 2 = a 2 , a 2 =a Notice that 2÷2=1. a 2 . We seek an expression whose square is  a 2 . Since   ( a ) 2 = a 2 , a 2 =a Notice that 2÷2=1.

Example 13

y 8 . We seek an expression whose square is  y 8 . Since   ( y 4 ) 2 = y 8 , y 8 = y 4 Notice that 8÷2=4. y 8 . We seek an expression whose square is  y 8 . Since   ( y 4 ) 2 = y 8 , y 8 = y 4 Notice that 8÷2=4.

Example 14

25 m 2 n 6 . We seek an expression whose square is 25 m 2 n 6 . Since   ( 5m n 3 ) 2 =25 m 2 n 6 , 25 m 2 n 6 =5m n 3 Notice that 2÷2=1 and 6÷2=3. 25 m 2 n 6 . We seek an expression whose square is 25 m 2 n 6 . Since   ( 5m n 3 ) 2 =25 m 2 n 6 , 25 m 2 n 6 =5m n 3 Notice that 2÷2=1 and 6÷2=3.

Example 15

− 121 a 10 ( b−1 ) 4 . We seek an expression whose square is 121 a 10 ( b−1 ) 4 . Since [ 11 a 5 ( b−1 ) 2 ] 2 = 121 a 10 ( b−1 ) 4 , 121 a 10 ( b−1 ) 4 = 11 a 5 ( b−1 ) 2 Then, − 121 a 10 ( b−1 ) 4 = −11 a 5 ( b−1 ) 2 Notice that 10÷2=5 and 4÷2=2. − 121 a 10 ( b−1 ) 4 . We seek an expression whose square is 121 a 10 ( b−1 ) 4 . Since [ 11 a 5 ( b−1 ) 2 ] 2 = 121 a 10 ( b−1 ) 4 , 121 a 10 ( b−1 ) 4 = 11 a 5 ( b−1 ) 2 Then, − 121 a 10 ( b−1 ) 4 = −11 a 5 ( b−1 ) 2 Notice that 10÷2=5 and 4÷2=2.

Practice Set D

Simplify each expression by removing the radical sign. Assume each variable is nonnegative.

Exercise 18

y 8 y 8

Solution

y 4 y 4

Exercise 19

16 a 4 16 a 4

Solution

4 a 2 4 a 2

Exercise 20

49 x 4 y 6 49 x 4 y 6

Solution

7 x 2 y 3 7 x 2 y 3

Exercise 21

− 100 x 8 y 12 z 2 − 100 x 8 y 12 z 2

Solution

−10 x 4 y 6 z −10 x 4 y 6 z

Exercise 22

− 36 ( a+5 ) 4 − 36 ( a+5 ) 4

Solution

−6 ( a+5 ) 2 −6 ( a+5 ) 2

Exercise 23

225 w 4 ( z 2 −1 ) 2 225 w 4 ( z 2 −1 ) 2

Solution

15 w 2 ( z 2 −1 ) 15 w 2 ( z 2 −1 )

Exercise 24

0.25 y 6 z 14 0.25 y 6 z 14

Solution

0.5 y 3 z 7 0.5 y 3 z 7

Exercise 25

x 2n , x 2n , where n n is a natural number.

Solution

x n x n

Exercise 26

x 4n , x 4n , where n n is a natural number.

Solution

x 2n x 2n

Exercises

Exercise 27

How many square roots does every positive real number have?

Solution

two

Exercise 28

The symbol represents which square root of a number?

Exercise 29

The symbol – represents which square root of a number?

Solution

secondary

For the following problems, find the two square roots of the given number.

Exercise 30

64

Exercise 31

81

Solution

9 and −9 −9

Exercise 32

25

Exercise 33

121

Solution

11 and −11 −11

Exercise 34

144

Exercise 35

225

Solution

15 and −15 −15

Exercise 36

10,000

Exercise 37

116116

Solution

1 4  and− 1 4 1 4  and− 1 4

Exercise 38

149149

Exercise 39

25362536

Solution

5 6  and − 5 6 5 6  and − 5 6

Exercise 40

121225121225

Exercise 41

0.04 0.04

Solution

0.2 and−0.2 0.2 and−0.2

Exercise 42

0.16 0.16

Exercise 43

1.21 1.21

Solution

1.1 and−1.1 1.1 and−1.1

For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number."

Exercise 44

4949

Exercise 46

− 36 − 36

Exercise 47

− 100 − 100

Solution

−10 −10

Exercise 48

− 169 − 169

Exercise 49

−3681−3681

Solution

− 2 3 − 2 3

Exercise 50

−121169−121169

Exercise 51

−225−225

Solution

not a real number

Exercise 52

−36−36

Exercise 53

−−1−−1

Solution

not a real number

Exercise 54

−−5−−5

Exercise 55

−( − 9 ) −( − 9 )

Solution

3

Exercise 56

−(−0.81)−(−0.81)

For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number.

Exercise 57

y+10y+10

Solution

y≥−10 y≥−10

Exercise 58

x+4x+4

Exercise 59

a−16a−16

Solution

a≥16 a≥16

Exercise 60

h−11h−11

Exercise 61

2k−12k−1

Solution

k≥ 1 2 k≥ 1 2

Exercise 62

7x+87x+8

Exercise 63

−2x−8−2x−8

Solution

x≤−4 x≤−4

Exercise 64

−5y+15−5y+15

For the following problems, simplify each expression by removing the radical sign.

Exercise 65

m6m6

Solution

m 3 m 3

Exercise 66

k10k10

Exercise 67

a8a8

Solution

a 4 a 4

Exercise 68

h16h16

Exercise 69

x4y10x4y10

Solution

x 2 y 5 x 2 y 5

Exercise 70

a6b20a6b20

Exercise 71

a4b6a4b6

Solution

a 2 b 3 a 2 b 3

Exercise 72

x8y14x8y14

Exercise 73

81a2b281a2b2

Solution

9ab 9ab

Exercise 74

49x6y449x6y4

Exercise 75

100m8n2100m8n2

Solution

10 m 4 n 10 m 4 n

Exercise 76

225p14r16225p14r16

Exercise 77

36x22y4436x22y44

Solution

6 x 11 y 22 6 x 11 y 22

Exercise 78

169w4z6(m−1)2169w4z6(m−1)2

Exercise 79

25x12(y−1)425x12(y−1)4

Solution

5 x 6 ( y−1 ) 2 5 x 6 ( y−1 ) 2

Exercise 80

64a10(a+4)1464a10(a+4)14

Exercise 81

9m6n4(m+n)189m6n4(m+n)18

Solution

3 m 3 n 2 ( m+n ) 9 3 m 3 n 2 ( m+n ) 9

Exercise 82

25m26n42r66s8425m26n42r66s84

Exercise 83

(f−2)2(g+6)4(f−2)2(g+6)4

Solution

( f−2 ) ( g+6 ) 4 ( f−2 ) ( g+6 ) 4

Exercise 84

(2c−3)6+(5c+1)2(2c−3)6+(5c+1)2

Exercise 85

−64r4s22−64r4s22

Solution

−8 r 2 s 11 −8 r 2 s 11

Exercise 86

−121a6(a−4)8−121a6(a−4)8

Exercise 87

−[ −(w+6)2 ]−[ −(w+6)2 ]

Solution

w+6 w+6

Exercise 88

−[ −4a2b2(c2+8)2 ]−[ −4a2b2(c2+8)2 ]

Exercise 89

1.21h4k4 1.21h4k4

Solution

1.1 h 2 k 2 1.1 h 2 k 2

Exercise 90

2.25m6p62.25m6p6

Exercise 91

−169a2b4c6196x4y6z8−169a2b4c6196x4y6z8

Solution

− 13a b 2 c 3 14 x 2 y 3 z 4 − 13a b 2 c 3 14 x 2 y 3 z 4

Exercise 92

−[ 81y4(z−1)2225x8z4w6 ]−[ 81y4(z−1)2225x8z4w6 ]

Exercised for Review

Exercise 93

((Reference)) Find the quotient. x2−14x2−1÷x−12x+1.x2−14x2−1÷x−12x+1.

Solution

x+1 2x−1 x+1 2x−1

Exercise 94

((Reference)) Find the sum. 1 x+1 + 3 x+1 + 2 x 2 −1 . 1 x+1 + 3 x+1 + 2 x 2 −1 .

Exercise 95

((Reference)) Solve the equation, if possible: 1x−2=3x2−x−2−3x+1.1x−2=3x2−x−2−3x+1.

Solution

No solution ; x=2 ; x=2 is excluded.

Exercise 96

((Reference)) Perform the division: 15 x 3 −5 x 2 +10x 5x . 15 x 3 −5 x 2 +10x 5x .

Exercise 97

((Reference)) Perform the division: x3−5x2+13x−21x−3.x3−5x2+13x−21x−3.

Solution

x 2 −2x+7 x 2 −2x+7

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