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Exercise Supplement

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. This module contains an exercise supplement for the chapter "Roots, Radicals, and Square Root Equations".

Exercise Supplement

Square Root Expressions ((Reference)) - Addition and Subtraction of Square Root Expressions ((Reference))

For the following problems, simplify the expressions.

Exercise 1

10 2 10 2

Solution

2 5 2 5

Exercise 2

6 8 6 8

Exercise 3

18 40 18 40

Solution

12 5 12 5

Exercise 4

11 11 11 11

Exercise 5

y y y y

Solution

y y

Exercise 6

r 3 r 3 r 3 r 3

Exercise 7

m+3 m+3 m+3 m+3

Solution

m+3 m+3

Exercise 8

a−7 a−7 a−7 a−7

Exercise 9

x 2 +4x+4 x 2 +4x+4

Solution

x+2 x+2

Exercise 10

y 2 −12y+36 y 2 −12y+36

Exercise 11

x+5 x+2 x+5 x+2

Solution

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

Exercise 12

n−3 n−1 n−3 n−1

Exercise 13

50 2 50 2

Solution

5

Exercise 14

75 5 3 75 5 3

Exercise 15

a 2 +6a+9 a+3 a 2 +6a+9 a+3

Solution

a+3 a+3

Exercise 16

4 x 2 +4x+1 2x+1 4 x 2 +4x+1 2x+1

Exercise 17

x 2 −11x+24 x−8 x 2 −11x+24 x−8

Solution

x−3 x−3

Exercise 18

y 2 +11y+28 y+4 y 2 +11y+28 y+4

Exercise 19

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

Solution

3+ 15 3+ 15

Exercise 20

5 ( 6 − 10 ) 5 ( 6 − 10 )

Exercise 21

a ( a − bc ) a ( a − bc )

Solution

a− abc a− abc

Exercise 22

x ( x 5 − 3x ) x ( x 5 − 3x )

Exercise 23

7 a 3 ( 2a − 4 a 3 ) 7 a 3 ( 2a − 4 a 3 )

Solution

a 2 14 −2 a 3 7 a 2 14 −2 a 3 7

Exercise 24

3 7 3 7

Exercise 25

2 5 2 5

Solution

2 5 5 2 5 5

Exercise 26

6 2 6 2

Exercise 27

8y y 8y y

Solution

8 y 8 y

Exercise 28

16 a 2 5a 16 a 2 5a

Exercise 29

( 2+ 3 )( 2− 3 ) ( 2+ 3 )( 2− 3 )

Solution

1

Exercise 30

( x+ 8 )( 3x+ 8 ) ( x+ 8 )( 3x+ 8 )

Exercise 31

( 4y− 3x )( 4y+ 3x ) ( 4y− 3x )( 4y+ 3x )

Solution

16 y 2 −3x 16 y 2 −3x

Exercise 32

( 6r+ 2s )( 4r+ 2s ) ( 6r+ 2s )( 4r+ 2s )

Exercise 33

2 2+ 7 2 2+ 7

Solution

− 2( 2− 7 ) 3 − 2( 2− 7 ) 3

Exercise 34

4 1− 6 4 1− 6

Exercise 35

6 x+ y 6 x+ y

Solution

6( x− y ) x 2 −y 6( x− y ) x 2 −y

Exercise 36

10 a− 2b 10 a− 2b

Exercise 37

5 a+ 3 5 a+ 3

Solution

a 5 − 15 a 2 −3 a 5 − 15 a 2 −3

Exercise 38

2 1+ 10 2 1+ 10

Exercise 39

8+ 3 2+ 6 8+ 3 2+ 6

Solution

8 6 −2 3 +3 2 −16 2 8 6 −2 3 +3 2 −16 2

Exercise 40

4+ 11 4− 11 4+ 11 4− 11

Exercise 41

36 a 4 b 5 c 11 x 2 y 5 36 a 4 b 5 c 11 x 2 y 5

Solution

6 a 2 b 2 c 5 bcy x y 3 6 a 2 b 2 c 5 bcy x y 3

Exercise 42

x 12 y 10 z 8 w 7 x 12 y 10 z 8 w 7

Exercise 43

32 x 5 y ( x−2 ) 3 32 x 5 y ( x−2 ) 3

Solution

4 x 2 ( x−2 ) 2xy( x−2 ) 4 x 2 ( x−2 ) 2xy( x−2 )

Exercise 44

−2 60 r 4 s 3 −2 60 r 4 s 3

Exercise 45

3 16 3 16

Solution

3 4 3 4

Exercise 46

4 25 4 25

Exercise 47

9 16 9 16

Solution

3 4 3 4

Exercise 48

5 36 5 36

Exercise 49

1 6 1 6

Solution

6 6 6 6

Exercise 50

3 10 3 10

Exercise 51

( x+4 ) 4 ( x−1 ) 5 ( x+4 ) 4 ( x−1 ) 5

Solution

( x+4 ) 2 ( x−1 ) 2 ( x−1 ) ( x+4 ) 2 ( x−1 ) 2 ( x−1 )

Exercise 52

( 3x+5 ) 3 ( 2x−7 ) 3 ( 3x+5 ) 3 ( 2x−7 ) 3

Exercise 53

( y−3z ) 12 ( y+3z ) 10 ( y−5z ) 3 ( y−3z ) 12 ( y+3z ) 10 ( y−5z ) 3

Solution

( y−3z ) 6 ( y+3z ) 5 ( y−5z ) y−5z ( y−3z ) 6 ( y+3z ) 5 ( y−5z ) y−5z

Exercise 54

( 8a−5b ) 26 ( 2a−9b ) 40 ( a−b ) 15 ( 8a−5b ) 26 ( 2a−9b ) 40 ( a−b ) 15

Exercise 55

4 11 +8 11 4 11 +8 11

Solution

12 11 12 11

Exercise 56

− 6 +5 6 − 6 +5 6

Exercise 57

5 60 −7 15 5 60 −7 15

Solution

3 15 3 15

Exercise 58

4a x 2 75 x 4 +6a 3 x 8 4a x 2 75 x 4 +6a 3 x 8

Exercise 59

−3 54 −16 96 −3 54 −16 96

Solution

−73 6 −73 6

Exercise 60

18 x 2 y 2 x 2 y 18 x 2 y 2 x 2 y

Exercise 61

4 x 2 +32x+64 + 10 x 2 +80x+160 4 x 2 +32x+64 + 10 x 2 +80x+160

Solution

( 2+ 10 )( x+4 ) ( 2+ 10 )( x+4 )

Exercise 62

−2 9 x 2 −42x+49 +5 18 x 2 −84x+98 −2 9 x 2 −42x+49 +5 18 x 2 −84x+98

Exercise 63

−10 56 a 3 b 7 +2 a 2 b 126a b 5 −10 56 a 3 b 7 +2 a 2 b 126a b 5

Solution

( −20a b 3 +6 a 2 b 3 ) 14ab ( −20a b 3 +6 a 2 b 3 ) 14ab

Exercise 64

3x − 5x 7x + 2x 3x − 5x 7x + 2x

Exercise 65

6a + 2a 3a − 5a 6a + 2a 3a − 5a

Solution

−3 2 − 30 − 6 − 10 2 −3 2 − 30 − 6 − 10 2

Square Root Equations with Applications ((Reference))

For the following problems, solve the equations.

Exercise 66

3x =9 3x =9

Exercise 67

4a =16 4a =16

Solution

a=64 a=64

Exercise 68

x +7=4 x +7=4

Exercise 69

a+6 =−5 a+6 =−5

Solution

no solution

Exercise 70

4a+5 =21 4a+5 =21

Exercise 71

3m+7 =10 3m+7 =10

Solution

m=31 m=31

Exercise 72

y+10 =5 y+10 =5

Exercise 73

a−7 =6 a−7 =6

Solution

a=43 a=43

Exercise 74

4x−8 =x−2 4x−8 =x−2

Exercise 75

2x+3 +8=11 2x+3 +8=11

Solution

x=3 x=3

Exercise 76

a 2 +5 +5=a a 2 +5 +5=a

Exercise 77

5b+4 −5=−2 5b+4 −5=−2

Solution

b=1 b=1

Exercise 78

2a+1 −10=−3 2a+1 −10=−3

Exercise 79

2x+5 = x+3 2x+5 = x+3

Solution

x=−2 x=−2

Exercise 80

5a−11 5a−11 = 2a−16

Exercise 81

At a small business, the monthly number of sales S S is approximately related to the number of employees E E by S=140+8 E−2 . S=140+8 E−2 .

(a) Determine the approximate number of sales if the number of employees is 27.

(b) Determine the approximate number of employees if the monthly sales are 268.

Solution

( a ) S=180;  ( b ) E=258 ( a ) S=180;  ( b ) E=258

Exercise 82

The resonance frequency f f in an electronic circuit containing inductance L L and capacitance C C in series is given by

 f= 1 2π LC f= 1 2π LC

(a) Determine the resonance frequency in an electronic circuit if the inductance is 9 and the capacitance is 0.0001 0.0001 . Use π=3.14. π=3.14.

(b) Determine the inductance in an electric circuit if the resonance frequency is 5.308 5.308 and the capacitance is 0.0001 0.0001 . Use π=3.14. π=3.14.

Exercise 83

If two magnetic poles of strength m m and m ' m ' units are at a distance r r centimeters (cm) apart, the force F F of repulsion in air between them is given by

 F= m m ' r 2 F= m m ' r 2

(a) Determine the force of repulsion if two magnetic poles of strengths 22 and 46 units are 8 cm apart.

(b) Determine how far apart are two magnetic poles of strengths 14 and 16 units if the force of repulsion in air between them is 42 units.

Solution

( a ) F=15.8125  ( b ) r=12.31 cm ( a ) F=15.8125  ( b ) r=12.31 cm

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