chapter 1 — real numbers & expressions

Chapter 9 — Radical Expressions and Equations Caspers

Chapter 9 — Radical Expressions and Equations

Section 9.1 — Introduction to Square Roots — Part I

Section 9.2 — Simplifying Radical Expressions

Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions

Section 9.4 — Division of Radical Expressions

Section 9.5 — Simplifying Radical Expressions — Part II

Section 9.6 — Radical Equations

Answers

Math 154 ::

Elementary Algebra

Math 154 ::

Elementary Algebra

http://cabrillo.edu/~mcaspers/math154/book/answers/ch9.pdf

http://cabrillo.edu/~mcaspers/math154/book/answers/ch9.pdf

Math 154 :: Elementary Algebra Chapter 9 — Radical Expressions and Equations

Section 9.1 — Introduction to Square Roots 1 Caspers

Section 9.1 Introduction to Square Roots

Example:

Simplify. If the expression is not a real number, state so.

a) 4 9

4

To simplify a square root of a perfect square, use the fact that:

a b if and only if 2a b .

If a is nonnegative, then 2a a . If a is negative, then 2a a

4 9

4

27

12

71

2

7

2

Homework

1. In your own words, define the square root of a number.

2. In your own words, define the square of a number.

3. Does the square root of a negative number have a real number value?

4. What is the expression under the square root sign called?

Simplify. If the expression is not a real number, state so.

5. 36

6. 1

7. 144

8. 254

9. 100

10. 0

11. 4

12. 196

13. 149

14. 81

15. 964

16. 225

17. 169

18. 400

19. 1900

20. 16 4

21. 25 9


Section 9.1 — Introduction to Square Roots 2 Caspers

22. Simplify each of the following.

a) 2

4

b) 2

16

c) 2x if x is a nonnegative value.

d) 2

4

e) 2

9

f) 2x

if x is a negative value.

g) 2

4

h) 2

9

i) 2

x


Section 9.2 — Simplifying Radical Expressions — Part I 3 Caspers

4 12 720 15a c a c

Section 9.2 Simplifying Radical Expressions — Part I

For all problems in the remainder of Chapter 9, assume that all variables are nonnegative.

Examples:

Simplify. Assume that all variables are nonnegative. If the expression is not a real number, state so.

a) 10 91400x y

One way to simplify a radical expression is to factor the radicand into primes.

10 92 2 2 5 5 7 x y In this problem, there are a pair of 2s and a pair of 5s. Taking the square root of these

“pairs”, will result in the following:

10 92 5 2 7 x y This problem may also be viewed as 10 91000 14 x y .

Now work on the variables. To take the square root of a variable raised to a specific exponent, you may divide the

EXPONENT by 2.

5 42 5 2 7x y y Now, simplify by multiply the “outside” together and the “inside” together.

5 410 14x y y

b) 4 12 720 15a c a c

The easiest way to simplify a product of two square roots is to write the expression as the square root of a single product.

Before taking the square root or multiplying the constants, find the prime factorization of

each constant.

4 12 72 2 5 3 5 a a c c Multiply the variables together by adding their exponents.

16 82 2 5 5 3 a c This problem may also be viewed as 16 8100 3 a c .

8 42 5 3a c Now, simplify by multiply the “outside” together.

8 410 3a c

Homework

1. In your own words, describe how to simplify the square root of a non-perfect square.

2. When you “pull factors out from a square root”, what operation is between those factors and the factors that remain under the

square root?

3. If the radicand in a square root expression has a variable raised to an exponent, what is the short cut rule for simplifying the

square root for that variable?

4. In your own words, describe how simplifying each of the following expressions are different: 16

and 16x .



5. Determine whether each statement is true or false.

a) The square root of a positive perfect square is a rational number.

b) A rational number is always real.

c) A real number is always rational.

d) An irrational number is not real.

e) The square root of a positive non-perfect square is a real number.

f) The square root of a positive non-perfect square is an irrational number.

g) The square root of a negative perfect square is a real number.

h) An irrational number is real.

i) The square root of a negative non-perfect square is a real number.

6. When multiplying two single-term square root expressions, what’s the “easiest” step to take first?

7. Assuming all variables are nonnegative, what is the simplified answer for a problem that “squares a square root” or “square

roots a square”? In other words, what is the relationship between “squaring” and “square rooting”?

Simplify. Assume that all variables are nonnegative. If the expression is not a real number, state so.

8. 18

9. 20

10. 45

11. 200

12. 50

13. 162

14. 288

15. 112

16. 1575

17. 363

18. 432

19. 3 44

20. 5 12

21. 10 28

22. 7 54

23. 3x

24. 11z

25. 9k

26. 400n

27. 401p



28. 45y

29. 88a

30. 1616m

31. 12 998c d

32. 25 10108y z

33. 6 9192p q

34. 20 17 2320a c d

35. 21 32648v u

36. 88mn

37. 50 42490x y z

38. 150150a

39. 36 25540c d

Simplify. Assume that all variables are nonnegative. Note that there is no addition or subtraction in any of the following problems.

40. 5 5

41. 16 16

42. 2

8

43. 2

11

44. 2 6

45. 15 12

46. 18 42

47. 1 22

48. 20 45

49. 35 14

50. 63 28

51. 33 3x x

52. 25 10y y

53. 3 56 15a a

54. 5 427 6z z

55. 3 340 15m n mn



56. 4 38 2xy z xy

57. 3 510 35a c ac

58. 6 6 748 24p q p q

59. 2

625w

60. 2

511k

61. 2

93a b

62. 2 2

8 109 4x x

63. 2 2

32 7y z

64. 2 2

5 3m m

65. 2 2

4 4cd cd


Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions 7 Caspers

Section 9.3 Addition, Subtraction, and Multiplication of Radical Expressions

Examples:

Simplify.

a) 45 2 45 7 5

To add square roots, the radicands must be “like”. Before combining like-radicand terms, simplify each term by taking the

square root.

3 3 5 2 3 3 5 7 5 This problem may also be viewed as 9 5 2 9 5 7 5 .

3 5 2 3 5 7 5 Remember, the factor that is “pulled” out is multiplied by what remains in the

square root.

3 5 6 5 7 5 Combine like-radicand terms. The radicand remains the same; only the constant

in front of the radical changes.

4 5

Multiply. All answers must be given in simplified form.

b) 8 3 6 3 2 6

To multiply a two term square root expression by another expression, distribute.

8 3 6 3 2 6 In this problem, use the FOIL method.

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

8 3 3 8 2 3 6 6 3 2 6 6 You may leave the problem as written to simplify each square root or

rewrite as:

8 9 16 18 18 2 36 This may be written as: 8 9 16 9 2 9 2 2 36

8 3 16 3 2 3 2 2 6

24 48 2 3 2 12 Combine like-radicand terms only.

12 45 2

Homework

1. Can you add any two square root expressions? In your own words, explain why or why not.

2. In your own words, describe how to add two square root expressions.

Simplify.

3. 4 2 5 2

4. 8 11 3 11

5. 4 10 10

6. 8 5 8

7. 9 14 14

8. 12 5 3 6 5



9. 12 4 12 7 2

10. 75 12

11. 20 45

12. 108 3

13. 100 44

14. 2 63 175

15. 5 192 3 80

16. 2 150 6

17. 121 18

18. 225 4 27

19. 2 36 275

20. 5 162 34 3 8

Multiply. All answers must be given in simplified form.

21. 2 5 2

22. 1 3 7 5

23. 2 1 0 6

24. 1 1 9 2

25. 6 3 5 2

26. 4 8 3 1 2

27. 6x x

28. y y y

29. 2 3a a a

30. 5 3 5 4

31. 1 4 6 3 1 4

32. 5 1 1 8 3 1 1

33. 2 7 6 1 0 6

34. 1 0 3 2 4 5

35. 4 6 9 14 21



36. 8 1 3x x

37. 3 3 3a a a

38. 4 4 4z z z

39. 5 5 5c c c

40. 2 7 2 7 7n n n

41. 6 2 3 5

42. 8 3 1 7

43. 4 6 10 3

44. 11 2 5 2

45. 1 14 7 1 4

46. 1 0 3 2 4 2 5

47. 12 3 5 8 3

48. 8 6 2 6 7 2

49. 3 2 7 9 2 4 7

50. 2

4 1 0

51. 2

1 3

52. 2 3x x x x

53. 2

y z

54. 4 3 2 3n n n n

55. 3 2 5 3 2 5

56. 6 2 7 6 2 7

57. 8 8w w w w

58. 9 4 9 4a a a a

59. 3 10 3 10z z z z

60. In your own words, describe the similarities in the problems 55 through 59 above. In your own words, describe the patter

and what is common about all of the answers.


Section 9.4 — Division of Radical Expressions 10 Caspers

Section 9.4 Division of Radical Expressions

Examples:

Simplify. Assume all variables are nonnegative. You may need to rationalize the denominator in order to give a simplified answer.

a) 12 7

3 11

240

140

x y

x y

To simplify the division of two square root expressions, the easiest first step is to write the radicands under one square root

sign and then reduce or simplify the fraction.

12 7

3 11

240

140

x y

x y

9

4

12

7

x

y

Next, take the square root of the numerator and the square root of the denominator.

9

4

4 3

7

x

y

4

2

2 3

7

x x

y

If the above two steps result in an expression that does NOT have a radical in the denominator, the problem is simplified. If

there is still a radical in the denominator, rationalize the denominator.

For this problem, there is still a radical in the denominator, so rationalize the denominator by multiplying the fraction by 7

7

4

2

2 3 7

7 7

x x

y

4

2

2 21

7

x x

y

Rationalize the denominator.

b) 2 3

1 5

To rationalize the denominator of an expression that contains two terms in the denominator, multiply the numerator and

denominator by the conjugate of the denominator.

1 52 3

1 5 1 5

2 3 1 5

1 5

2 3 1 5

4

Notice how the denominator is multiplied (FOIL-ed) out, but the numerator is left in factored form.

In many cases, this will be the last step, but for this problem, the fraction can be simplified as it contains a factor of 3 in the

numerator and denominator.

2 3 1 5

4

2

3 1 5

2

Your answer can also be written as

3 15

2

.

Homework

1. When simplifying a square root whose radicand is a quotient, what is usually the “easiest” first step?

2. When looking at an expression that consists of a quotient of square roots (without an addition or subtraction), should you take the

square root first or simplify the fraction first?



3. What are the three conditions that must be met in order for a square root expression to be considered “simplified”?

4. In your own words, describe, in general what it means to rationalize a denominator.

5. What are the two different types of expressions you will see in this class where you might need to rationalize the denominator? In

your own words, describe how to rationalize the denominator for each type.

Simplify. Assume all variables are nonnegative.

6. 25

9

7. 75

3

8. 16

4

9. 18

2

10. 10

5

11. 24

3

12. 100

2

13. 18

72

14. 45

5

15. 105

21

16. 98

18

17. 110

90

18. 56

7

19. 7

3

x

x

20. 8

6

40

125

m

m

21. 8

6

28

63

a c

a c



22. 4

12

2

162

y

y

23. 10

5

78

24

wz

w z

24. 11 21

13 5

75

27

p q

p q

25. 25 9

3 4

80

16

m n

m n

26. 4 8

6 12

3

192

x y z

x y z

27. 16 25

9

1815

15

c d

c d

Simplify. Assume all variables are nonnegative. You may need to rationalize the denominator in order to give a simplified answer.

28. 8

3

29. 7

2

30. 6

10

31. 12

15

32. 14

7

33. 7

14

34. 5

y

35. 2

3

4

20

a

a

36. 8

2 p

37. 8

11

x

x

38. 13

1018

m

m



39. 4

2

52

26

z

z

40. 15 10

2 16

180

640

c d

c d

41. 13 3

14 10

168

84

x y

x y

42. 4

2 6

810

2430

pq

p q

43. 17 3

6

33

88

m n

mn

44. 20

11

63

35

a c

a c

45. 3

4 8

120

180

xy

x y

Rationalize the denominator.

46. 5

2 3

47. 1

3 7

48. 2

1 6

49. 8

3 5

50. 4 10

3 2

51. 6 3

8 5

52. 8

x

x

53. 11

11 y

54. 2 a

a c

55. xy

x y


Section 9.5 — Simplifying Radical Expressions — Part II 14 Caspers

Section 9.5 Simplifying Radical Expressions — Part II

9.5 — Simplifying Radical Expressions — Part II Worksheet

Example:

Simplify.

a) 6 3 6 2 4

4

To simplify an expression like the one above, follow order of operation. The first step is to simplify the radicand.

6 1 2

4

Next, simplify the square root if possible. Finally, if the numerator and denominator contain a common FACTOR, cancel it.

6 4 3

4

6 2 3

4

2 3 3

4

2 3 3

4

2

3 3

2

Homework

1. In your own words, describe the order of operations in mathematics. What “level” is taking a square root?

2. In your own words, describe the steps for simplifying an expression of the type b c d

a

.

Simplify.

3. 3 9 4

2

4. 5 2 5 1 6

8

5. 6 3 6 4

2

6. 1 0 1 0 0 1 6

4

7. 7 4 9 1

2

8. 8 6 4 2 0

1 0

9. 4 1 6 2 4

4

10. 1 1 1 2 0

1 2

http://cabrillo.edu/~mcaspers/math154/worksheets/9_5.pdf



Section 9.6 — Radical Equations 15 Caspers

2 0

2

x

x

1 0

1

x

x

Section 9.6 Radical Equations

9.6 — Review of Radical Expressions and Equations Worksheet

Example:

Solve. If there is no solution, state so.

a) 4 3 5x x

To solve a radical equation first ISOLATE the radical expression.

4 3 5x x Subtract 4 from both sides of the equation.

3 1x x

To “undo” the square rooting, square both sides of the equation.

2 2

3 1x x Don’t forget to FOIL when necessary.

Continue to solve the remaining equation.

3 1 1x x x

23 2 1x x x Subtract x and 3 from both sides of the equation.

20 2x x Since this is a quadratic equation, solve by factoring.

0 2 1x x Since this is a quadratic equation, solve by factoring.

or

Check all solutions!

Check for 2x : Check for 1x :

4 2 3 ? 2 5

4 1 3 ? 1 5

4 1 ? 3

4 4 ? 1 5

4 1 ? 3

4 2 ? 6

5 3

6 6

Since –2 doesn’t work, the solution is only 1x .

Homework

1. Multiply each of the following. If you know how to spot the difference between these problems and compute them correctly, you’ll

have an easier time solving the equations in this section.

a) 2

2x

b) 2

2 x

c) 2

2x

d) 2

2x

2. In your own words, describe how to recognize a square root equation. What is the inverse of square rooting?




Section 9.6 — Radical Equations 16 Caspers

3. In your own words, describe how to solve a square root equation. Is it necessary to check your answers when solving a square

root equation? In your own words, explain why or why not.

Solve. Check your answers. If there is no solution, state so.

4. 8x

5. 4y

6. 3 12a

7. 22 12 m

8. 2 1 3w

9. 1 4 11p

10. 7 13 3x

11. 3 5n

12. 10 6q

13. 17 5 23k

14. 13 8 3 12w

15. 33 35 7d

16. 45 2 6 49y

17. 3 8x x

18. 2 5 1 0k k

19. 1 5z z

20. 2 3 5 2a a

21. 7 38 3 4p p

22. 4 5 35n n

23. 7 3 3q q

24. 12x x

25. 8 6k k

26. 20w w

27. 7 13y y

28. 1 1z z

29. 2 3a a

30. 2 7 8 3c c

31. 2 30 14m m

32. 2 4 5p p

33. 6 2 3x x

chapter 1 — real numbers & expressions

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