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Math 154 :: Elementary Algebra 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 Chapter 9 — Radical Expressions and Equations Caspers Math 154 :: Elementary Algebra Section 9.1 Chapter 9 — Radical Expressions and Equations Introduction to Square Roots Example: Simplify. If the expression is not a real number, state so. 49 a) 4 To simplify a square root of a perfect square, use the fact that: a b if and only if a b2 . If a is nonnegative, then 49 4 a2 a . 7 1 2 If a is negative, then 2 1 7 2 a 2 a 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. 25 4 9. 100 10. 0 11. 4 12. 196 13. 1 49 14. 81 15. 9 64 16. 225 17. 169 18. 400 19. 1 900 20. 16 4 21. 25 9 Section 9.1 — Introduction to Square Roots Caspers 1 Math 154 :: Elementary Algebra Chapter 9 — Radical Expressions and Equations 22. Simplify each of the following. a) 4 2 b) 16 2 c) x2 d) 4 2 e) 9 2 f) x2 g) 4 h) 9 i) x if x is a nonnegative value. if x is a negative value. 2 2 2 Section 9.1 — Introduction to Square Roots Caspers 2 Math 154 :: Elementary Algebra Section 9.2 Chapter 9 — Radical Expressions and Equations 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) 1400x10 y9 One way to simplify a radical expression is to factor the radicand into primes. 2 2 2 5 5 7 x10 y9 2 5 2 7 x10 y9 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: This problem may also be viewed as 1000 14 x10 y9 . 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. 2 5 x5 y 4 2 7 y Now, simplify by multiply the “outside” together and the “inside” together. 10 x5 y 4 14 y b) 20a 4 c 15a12c7 The easiest way to simplify a product of two square roots is to write the expression as the square root of a single product. 20a 4c 15a12c7 Before taking the square root or multiplying the constants, find the prime factorization of each constant. 2 2 5 3 5 a4 a12 c c7 Multiply the variables together by adding their exponents. 2 2 5 5 3 a16 c8 This problem may also be viewed as 100 3 a16 c8 . 2 5 a8 c4 3 Now, simplify by multiply the “outside” together. 10a8c4 3 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 x16 . Section 9.2 — Simplifying Radical Expressions — Part I Caspers 3 Math 154 :: Elementary Algebra 5. Chapter 9 — Radical Expressions and Equations 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. x3 24. z11 25. k9 26. n400 27. p 401 Section 9.2 — Simplifying Radical Expressions — Part I Caspers 4 Math 154 :: Elementary Algebra 28. 5y 4 29. 8a8 30. 16m16 31. 98c12 d 9 32. 108y 25 z10 33. 192 p6 q9 34. 320a 20 c17 d 2 35. 648v 21u 32 36. 8mn8 37. 490x50 y 42 z 38. 150a150 39. 540c36 d 25 Chapter 9 — Radical Expressions and Equations 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. 8 43. 11 44. 2 6 45. 15 12 46. 18 42 47. 1 22 48. 20 45 49. 35 14 50. 63 28 51. 3x 52. 5 y 2 10 y 53. 6a3 15a5 54. 27 z 5 6 z 4 55. 40m3n 15mn3 2 2 3 x3 Section 9.2 — Simplifying Radical Expressions — Part I Caspers 5 Math 154 :: Elementary Algebra 56. 8xy 4 z 2 xy3 57. 10a3c 35ac5 58. 48 p6 q 24 p6 q7 59. 60. 61. 25w6 11k 3a b Chapter 9 — Radical Expressions and Equations 2 2 5 9 2 62. 9 x8 4 x10 63. 2 y3 7z 64. 5m 3m 65. 4cd 2 2 2 2 2 2 2 4cd 2 Section 9.2 — Simplifying Radical Expressions — Part I Caspers 6 Math 154 :: Elementary Algebra Section 9.3 Chapter 9 — Radical Expressions and Equations 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. 335 2 335 7 5 This problem may also be viewed as 95 2 95 7 5 . 3 5 23 5 7 5 Remember, the factor that is “pulled” out is multiplied by what remains in the square root. 3 56 57 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 32 6 To multiply a two term square root expression by another expression, distribute. 8 3 6 8 3 3 8 33 32 6 In this problem, use the FOIL method. 8 32 6 8 2 36 8 9 16 18 83 16 3 2 24 48 2 12 45 2 6 3 63 18 3 2 2 66 2 36 3 2 62 6 You may leave the problem as written to simplify each square root or rewrite as: This may be written as: 8 9 16 9 2 92 2 36 26 12 Combine like-radicand terms only. 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. 85 8 7. 9 14 14 8. 12 5 3 6 5 Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions Caspers 7 Math 154 :: Elementary Algebra 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 Chapter 9 — Radical Expressions and Equations Multiply. All answers must be given in simplified form. 22. 13 23. 2 24. 7 5 2 5 2 21. 10 6 11 9 2 25. 6 3 52 26. 4 8 3 12 27. x x 6 28. y y 29. 2a 3 a a 30. 5 3 54 31. 14 6 3 14 32. 5 11 8 3 11 33. 2 7 6 10 6 y 34. 10 3 2 4 5 35. 4 6 9 14 21 Section 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions Caspers 8 Math 154 :: Elementary Algebra 36. x 8 x 13 37. 3 a 3a 3 a 38. 4z 4 z 4z 5c 5c 5 c 39. 40. 2 7n 2 7n 7n 41. 6 2 3 5 42. 8 3 1 7 43. 4 6 1 0 3 44. 1 1 2 5 2 45. 1 46. 1 0 3 2 4 2 5 47. 1 2 3 5 8 3 48. 8 6 2 49. 3 2 7 9 50. 4 51. 1 3 52. x 2 x 3x x 53. y z 54. n 4 55. 3 2 5 3 2 5 56. 57. w 8 w w 8 w 58. 9 59. 3z z 60. 14 7 10 Chapter 9 — Radical Expressions and Equations 14 67 2 24 7 2 2 2 3n 2n 62 7 a 4a 9 10 3n 62 7 a 4a 3z z 10 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 9.3 — Addition, Subtraction, and Multiplication of Radical Expressions Caspers 9 Math 154 :: Elementary Algebra Section 9.4 Chapter 9 — Radical Expressions and Equations 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) 240 x12 y 7 140 x3 y11 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. 240 x12 y 7 12 x9 140 x3 y11 7 y4 Next, take the square root of the numerator and the square root of the denominator. 4 3 x9 7 y4 2 x 4 3x y2 7 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 2 x 4 3x y2 7 7 7 7 7 2 x 4 21x 7 y2 Rationalize the denominator. 2 3 b) 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 5 5 1 5 2 3 1 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 42 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? Section 9.4 — Division of Radical Expressions Caspers 10 Math 154 :: Elementary Algebra Chapter 9 — Radical Expressions and Equations 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. 25 6. 9 7. 75 3 8. 16 4 9. 18 2 10. 10 5 11. 24 3 12. 100 2 13. 18 72 14. 15. 16. 17. 18. 19. 20. 21. 45 5 105 21 98 18 110 90 56 7 x7 x3 40m8 125m6 28a8 c 63a 6 c Section 9.4 — Division of Radical Expressions Caspers 11 Math 154 :: Elementary Algebra 22. 23. 24. 25. 26. 27. Chapter 9 — Radical Expressions and Equations 2 y4 162 y12 78wz10 24w5 z 75 p11q 21 27 p13 q5 80m25 n9 16m3 n 4 3x 4 y8 z 192 x6 y12 z 1815c16 d 25 15c9 d Simplify. Assume all variables are nonnegative. You may need to rationalize the denominator in order to give a simplified answer. 8 28. 3 29. 7 2 30. 6 10 31. 12 15 32. 33. 34. 35. 36. 37. 38. 14 7 7 14 5 y 4a 2 20a 3 8 2p x8 x11 m13 18m10 Section 9.4 — Division of Radical Expressions Caspers 12 Math 154 :: Elementary Algebra 39. 40. 41. 42. 43. 44. 45. Chapter 9 — Radical Expressions and Equations 52 z 4 26 z 2 180c15 d 10 640c 2 d 16 168 x13 y 3 84 x14 y10 810 pq 4 2430 p 2 q 6 33m17 n3 88mn6 63a 20 c 35a11c 120 xy 3 180 x 4 y8 Rationalize the denominator. 5 46. 2 3 47. 48. 49. 50. 51. 52. 53. 54. 55. 1 3 7 2 1 6 8 3 5 4 10 3 2 6 3 8 5 x 8 x 11 11 y 2 a a c xy x y Section 9.4 — Division of Radical Expressions Caspers 13 Math 154 :: Elementary Algebra Section 9.5 Chapter 9 — Radical Expressions and Equations Simplifying Radical Expressions — Part II 9.5 — Simplifying Radical Expressions — Part II Worksheet Example: Simplify. 6 36 24 a) 4 To simplify an expression like the one above, follow order of operation. The first step is to simplify the radicand. 6 12 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 42 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 25 16 8 5. 6 36 4 2 6. 10 100 16 4 7. 7 49 1 2 8. 8 64 20 10 9. 4 16 24 4 10. 1 1 120 12 Section 9.5 — Simplifying Radical Expressions — Part II Caspers 14 Math 154 :: Elementary Algebra Section 9.6 Chapter 9 — Radical Expressions and Equations Radical Equations 9.6 — Review of Radical Expressions and Equations Worksheet Example: Solve. If there is no solution, state so. a) 4 x 3 x 5 To solve a radical equation first ISOLATE the radical expression. 4 x3 x5 Subtract 4 from both sides of the equation. x 3 x 1 To “undo” the square rooting, square both sides of the equation. x3 2 x 1 2 Don’t forget to FOIL when necessary. Continue to solve the remaining equation. x 3 x 1 x 1 x 3 x2 2 x 1 Subtract x and 3 from both sides of the equation. 0 x2 x 2 Since this is a quadratic equation, solve by factoring. 0 x 2 x 1 Since this is a quadratic equation, solve by factoring. x 1 0 x20 or 2 x x 1 Check all solutions! Check for x 1 : Check for x 2 : 4 2 3 ? 2 5 4 1 ? 3 41 ? 3 5 3 4 1 3 ? 1 5 4 4 ? 1 5 4 2 ? 6 6 6 Since –2 doesn’t work, the solution is only x 1 . Homework 1. 2. 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) 2x 2 b) 2 x c) x 2 2 d) 2 x2 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 Caspers 15 Math 154 :: Elementary Algebra 3. Chapter 9 — Radical Expressions and Equations 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. x 8 5. y 4 6. 3 a 12 7. 22 12 m 8. 2w 1 3 9. 1 4 p 11 10. 7 1 3 3x 11. n3 5 12. 10 q6 13. 17 k 5 23 14. 1 3 8w 3 1 2 15. 33 35 d 7 16. 4 5 2y 6 4 9 17. 3x 8 18. 2k 5 k 1 0 19. z 1 20. 2 3a 5 21. 22. x z5 2a 7p 38 3 p 4 4 n5 n 35 23. 7q 3 3 q 24. x x 12 25. k 8 k 6 26. w w 20 27. 7 28. 1 29. 2 a a3 30. 2 7c 8 3c 31. 2m 3 0m 1 4 32. 2 4p 5 p 33. 6 2x 3 x y 13 y z 1 z Section 9.6 — Radical Equations Caspers 16