advertisement

10-2 Simplifying Radical Expressions Simplify each expression. 1. SOLUTION: 2. SOLUTION: 3. SOLUTION: 4. SOLUTION: 5. SOLUTION: eSolutions Manual - Powered by Cognero Page 1 10-2 Simplifying Radical Expressions 5. SOLUTION: 6. SOLUTION: 7. SOLUTION: 8. SOLUTION: 9. SOLUTION: eSolutions Manual - Powered by Cognero Page 2 10-2 Simplifying Radical Expressions 9. SOLUTION: 10. MULTIPLE CHOICE Which expression is equivalent to ? A B C D SOLUTION: The correct choice is D. Simplify each expression. 11. eSolutions Manual - Powered by Cognero SOLUTION: Page 3 10-2 Simplifying Radical Expressions The correct choice is D. Simplify each expression. 11. SOLUTION: 12. SOLUTION: 13. SOLUTION: eSolutions Manual - Powered by Cognero Page 4 10-2 Simplifying Radical Expressions 13. SOLUTION: 14. SOLUTION: 15. SOLUTION: eSolutions Manual - Powered by Cognero Page 5 10-2 Simplifying Radical Expressions 15. SOLUTION: 16. SOLUTION: Simplify each expression. 17. SOLUTION: 18. eSolutions Manual - Powered by Cognero SOLUTION: Page 6 10-2 Simplifying Radical Expressions 18. SOLUTION: 19. SOLUTION: 20. SOLUTION: 21. SOLUTION: 22. eSolutions Manual - Powered by Cognero SOLUTION: Page 7 10-2 Simplifying Radical Expressions 22. SOLUTION: 23. SOLUTION: 24. SOLUTION: 25. SOLUTION: 26. SOLUTION: eSolutions Manual - Powered by Cognero Page 8 10-2 Simplifying Radical Expressions 26. SOLUTION: 27. SOLUTION: 28. SOLUTION: 29. SOLUTION: 30. SOLUTION: eSolutions Manual - Powered by Cognero Page 9 10-2 Simplifying Radical Expressions 30. SOLUTION: 31. SOLUTION: 32. SOLUTION: 33. SOLUTION: 34. SOLUTION: eSolutions Manual - Powered by Cognero Page 10 10-2 Simplifying Radical Expressions 34. SOLUTION: 35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by , where h is the height of the hill in feet. a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill. SOLUTION: a. b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the equation . The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill. 36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is modeled by the function , where h represents the maximum height of the water and g represents the acceleration due to 2 gravity (32 ft/s ). a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pump advertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will a pump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain. SOLUTION: a. eSolutions Manual - Powered by Cognero Page 11 10-2 Simplifying Radical Expressions The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill. 36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is modeled by the function , where h represents the maximum height of the water and g represents the acceleration due to 2 gravity (32 ft/s ). a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pump advertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will a pump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain. SOLUTION: a. b. To determine the height of the water, substitute 70 for v in the function . A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77 for v in the function . A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need. Simplify each expression. 37. SOLUTION: eSolutions Manual - Powered by Cognero Page 12 10-2ASimplifying pump with aRadical velocityExpressions of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need. Simplify each expression. 37. SOLUTION: 38. SOLUTION: 39. eSolutions Manual - Powered by Cognero SOLUTION: Page 13 10-2 Simplifying Radical Expressions 39. SOLUTION: 40. SOLUTION: 41. SOLUTION: eSolutions Manual - Powered by Cognero Page 14 10-2 Simplifying Radical Expressions 41. SOLUTION: 42. SOLUTION: eSolutions Manual - Powered by Cognero Page 15 10-2 Simplifying Radical Expressions 42. SOLUTION: 43. SOLUTION: 44. SOLUTION: eSolutions Manual - Powered by Cognero Page 16 10-2 Simplifying Radical Expressions 44. SOLUTION: 45. SOLUTION: 46. SOLUTION: eSolutions Manual - Powered by Cognero Page 17 10-2 Simplifying Radical Expressions 46. SOLUTION: 47. SOLUTION: 48. SOLUTION: eSolutions Manual - Powered by Cognero Page 18 10-2 Simplifying Radical Expressions 48. SOLUTION: 49. ELECTRICITY The amount of current in amperes I that an appliance uses can be calculated using the formula , where P is the power in watts and R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms? SOLUTION: a. b. Substitute P = 75 and R = 5 in the equation . An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms. 50. KINETIC ENERGY The speed eSolutions Manual - Powered by Cognero v of a ball can be determined by the equation energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. , where k is the kinetic Page 19 10-2 Simplifying Radical Expressions An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms. 50. KINETIC ENERGY The speed v of a ball can be determined by the equation , where k is the kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules? SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation. b. Substitute V = 7 in the equation . The kinetic energy of the ball is 73.5 Joules. 51. SUBMARINES The greatest distance d in miles that the lookout can see on a clear day is modeled by the formula . Determine how high the submarine would have to raise its periscope to see a ship, if the submarine is the given distances away from a ship. eSolutions Manual - Powered by Cognero Page 20 10-2 Simplifying Radical Expressions The kinetic energy of the ball is 73.5 Joules. 51. SUBMARINES The greatest distance d in miles that the lookout can see on a clear day is modeled by the formula . Determine how high the submarine would have to raise its periscope to see a ship, if the submarine is the given distances away from a ship. SOLUTION: First solve the equation for h. Distance Height = 3 d 2 h= 6 2 (3) = 6 h = 9 2 (6) = 24 h= 12 2 (9) = 54 h= 15 2 (12) = 96 h= 2 (15) = 150 52. CCSS STRUCTURE Explain how to solve . SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable. eSolutions Manual - Powered by Cognero Page 21 Distance 3 2 6 2 Height = d h = (3) = 6 h = 10-2 Simplifying Radical Expressions 9 2 (6) = 24 h= 12 2 (9) = 54 h= 15 2 (12) = 96 h= 2 (15) = 150 52. CCSS STRUCTURE Explain how to solve . SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable. Use the conjugate of to rationalize the denominator. So, the solution is . 53. CHALLENGE Simplify each expression. a. b. c. SOLUTION: a. b. c. 54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the reciprocal to eSolutions Manual - Powered by Cognero get . What number did she start with? Write a formula to describe the process. Page 22 b. 10-2 Simplifying Radical Expressions c. 54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the reciprocal to get . What number did she start with? Write a formula to describe the process. SOLUTION: Let x = a number. 55. OPEN ENDED Write two binomials of the form and . Then find their product. SOLUTION: Two binomials of the form and are and . 56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by solving the 2 quadratic equation ax + bx + c = 0. (Hint: Begin by completing the square.) SOLUTION: eSolutions Manual - Powered by Cognero Page 23 Two binomials of the form and are and . 10-2 Simplifying Radical Expressions 56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by solving the 2 quadratic equation ax + bx + c = 0. (Hint: Begin by completing the square.) SOLUTION: 57. WRITING IN MATH Summarize how to write a radical expression in simplest form. SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following. eSolutions Manual - Powered by Cognero 58. Jerry’s electric bill is $23 less than his natural gas bill. The two bills are a total of $109. Which of the following Page 24 10-2 Simplifying Radical Expressions 57. WRITING IN MATH Summarize how to write a radical expression in simplest form. SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following. 58. Jerry’s electric bill is $23 less than his natural gas bill. The two bills are a total of $109. Which of the following equations can be used to find the amount of his natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109 SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109. So, the correct choice is D. 2 59. Solve a − 2a + 1 = 25. F −4, −6 G 4, −6 H −4, 6 J 4, 6 SOLUTION: Solve for a. eSolutions Manual - Powered by Cognero Page 25 Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109. 10-2 Simplifying Radical Expressions So, the correct choice is D. 2 59. Solve a − 2a + 1 = 25. F −4, −6 G 4, −6 H −4, 6 J 4, 6 SOLUTION: Solve for a. The roots are –4 and 6. So, the correct choice is H. 60. The expression is equivalent to which of the following? A B C D SOLUTION: So, the correct choice is C. 61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make? SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales. Miki made $507.50 last week. eSolutions Manual - Powered by Cognero Graph each function. Compare to the parent graph. State the domain and range. 62. Page 26 10-2 Simplifying Radical Expressions So, the correct choice is C. 61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make? SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales. Miki made $507.50 last week. Graph each function. Compare to the parent graph. State the domain and range. 62. SOLUTION: x 0 1 y 1 –1 2 ≈ 1.8 The parent function a vertical stretch of 1}. 3 ≈ 2.5 4 3 is multiplied by a value greater than 1 and is subtracted by the value 1, so the graph is followed by a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ - 63. SOLUTION: x 0 1 y 0 2 ≈ 0.7 The parent function eSolutions Manual - Powered by Cognero 3 ≈0 .9 4 1 is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical compression of . Another way to identify the compression is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. Page 27 is multiplied by a value greater than 1 and is subtracted by the value 1, so the graph is The parent function a vertical stretch of followed by a translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ 10-2 Simplifying Radical Expressions 1}. 63. SOLUTION: x 0 1 y 0 2 ≈ 0.7 The parent function 3 ≈0 .9 4 1 is multiplied by a value less than 1 and greater than 0, so the the graph is a vertical compression of . Another way to identify the compression is to notice that the y-values in the table are times the corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}. 64. SOLUTION: x –2 –1 y 0 2 0 ≈ 2.8 1 ≈ 3.5 2 4 This graph is the result of a vertical stretch of the graph of y = is {x|x ≥ –2}, and the range is {y|y ≥ 0}. followed by a translation 2 units left. The domain 65. SOLUTION: x 0 –1 y 0 –1 1 ≈ – 1.4 eSolutions Manual - Powered by Cognero 2 ≈ – 1.7 Page 28 graph is the result Expressions of a vertical stretch of the graph of y = 10-2This Simplifying Radical is {x|x ≥ –2}, and the range is {y|y ≥ 0}. followed by a translation 2 units left. The domain 65. SOLUTION: x 0 –1 y 0 –1 1 ≈ – 1.4 2 ≈ – 1.7 This graph is the result of a reflection across the x-axis of the graph of The domain is {x|x ≥ –1}, and the range is {y|y ≤ 0}. followed by a translation 1 unit left. 66. SOLUTION: x 3 4 y 0 –3 5 ≈ – 4.2 6 ≈ – 5.2 This graph is the result of a vertical stretch of the graph of followed by a reflection across the x-axis, and then a translation 3 units right. The domain is {x|x ≥ 3}, and the range is {y|y ≤ 0}. 67. SOLUTION: x 0 1 y 1 –1 2 ≈ – 1.8 eSolutions Manual - Powered by Cognero 3 ≈ – 2.5 Page 29 graph is the result Expressions of a vertical stretch of the graph of followed by a reflection across the x-axis, and 10-2This Simplifying Radical then a translation 3 units right. The domain is {x|x ≥ 3}, and the range is {y|y ≤ 0}. 67. SOLUTION: x 0 1 y 1 –1 2 ≈ – 1.8 3 ≈ – 2.5 is multiplied by a value less than 1 and is added to the value 1, so the graph is a vertical The parent function stretch of followed by a reflection across the x-axis, and then a translation 1 unit up. The domain is {x|x ≥ 0}, and the range is {y|y ≤ 1}. Determine the domain and range for each function. 68. f (x) = |2x − 5| SOLUTION: Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0. Make a table of values. x 0 f (x) 5 1 3 2 1 3 1 4 3 The domain is all real numbers, and the range is {y | y ≥ 0}. eSolutions Manual - Powered by Cognero 69. h(x) = |x − 1| SOLUTION: Page 30 is multiplied by a value less than 1 and is added to the value 1, so the graph is a vertical The parent function of followed by a reflection across the x-axis, and then a translation 1 unit up. The domain is {x|x ≥ 10-2stretch Simplifying Radical Expressions 0}, and the range is {y|y ≤ 1}. Determine the domain and range for each function. 68. f (x) = |2x − 5| SOLUTION: Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0. Make a table of values. x 0 f (x) 5 1 3 2 1 3 1 4 3 The domain is all real numbers, and the range is {y | y ≥ 0}. 69. h(x) = |x − 1| SOLUTION: Since h(x) cannot be negative, the minimum point of the graph is where h(x) = 0. Make a table of values. x –4 –2 h(x) 5 3 eSolutions Manual - Powered by Cognero 0 1 2 1 4 3 Page 31 The domain is all real numbers, and the range is {y | y ≥ 0}. 10-2 Simplifying Radical Expressions 69. h(x) = |x − 1| SOLUTION: Since h(x) cannot be negative, the minimum point of the graph is where h(x) = 0. Make a table of values. x –4 –2 h(x) 5 3 0 1 2 1 4 3 The domain is all real numbers, and the range is {y | y ≥ 0}. 70. SOLUTION: This is a piecewise-defined function. Make a table of values. Be sure to include the domain values for which the function changes. x -1 0 1 2 3 g(x) -2 -1 0 1 -5 Notice that both functions are linear. The domain is all real numbers, and the range is {y | y ≤ 1}. eSolutions Manual - Powered by Cognero Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 2 71. x − 25 = 0 Page 32 domain isRadical all real numbers, and the range is {y | y ≥ 0}. 10-2The Simplifying Expressions 70. SOLUTION: This is a piecewise-defined function. Make a table of values. Be sure to include the domain values for which the function changes. x -1 0 1 2 3 g(x) -2 -1 0 1 -5 Notice that both functions are linear. The domain is all real numbers, and the range is {y | y ≤ 1}. Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. 2 71. x − 25 = 0 SOLUTION: For this equation, a = 1, b = 0, and c = –25. The solutions are 5 and –5. 2 72. r + 25 = 0 SOLUTION: For this equation, a = 1, b = 0, and c = 25. eSolutions Manual - Powered by Cognero Page 33 10-2The Simplifying solutions Radical are 5 andExpressions –5. 2 72. r + 25 = 0 SOLUTION: For this equation, a = 1, b = 0, and c = 25. There are no real positive square roots of –100. Therefore, the solution to this equation is ø. 2 73. 4w + 100 = 40w SOLUTION: Rewrite the equation in standard form. For this equation, a = 4, b = –40, and c = 100. The solution is 5. 2 74. 2r + r − 14 = 0 SOLUTION: For this equation, a = 2, b = 1, and c = –14. eSolutions Manual - Powered by Cognero Page 34 10-2 Simplifying Radical Expressions The solution is 5. 2 74. 2r + r − 14 = 0 SOLUTION: For this equation, a = 2, b = 1, and c = –14. The solutions are –2.9 and 2.4. 2 75. 5v − 7v = 1 SOLUTION: Rewrite the equation in standard form. For this equation, a = 5, b = –7, and c = –1. The solutions are −0.1 and 1.5. 2 76. 11z − z = 3 SOLUTION: Rewrite the equation in standard form. For this equation, a = 11, b = –1, and c = –3. eSolutions Manual - Powered by Cognero Page 35 10-2 Simplifying Radical Expressions The solutions are −0.1 and 1.5. 2 76. 11z − z = 3 SOLUTION: Rewrite the equation in standard form. For this equation, a = 11, b = –1, and c = –3. The solutions are −0.5 and 0.6. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime . 2 77. n − 81 SOLUTION: 78. 4 − 9a 2 SOLUTION: 5 79. 2x − 98x 3 SOLUTION: 4 80. 32x − 2y 4 SOLUTION: eSolutions Manual - Powered by Cognero Page 36 SOLUTION: 10-2 Simplifying Radical Expressions 4 80. 32x − 2y 4 SOLUTION: 2 81. 4t − 27 SOLUTION: In this trinomial, a = 4, b = 0 and c = –27, so m + p is zero and mp is negative. Therefore, m and p must be opposite signs. List the factors of 4(–27) or –108 with a sum of 0. Factors of –108 Sum of 0 1, –108 –107 107 –1, 108 2, –52 –50 50 –2, 52 3, –36 –33 33 –3, 36 4, –27 –23 23 –4, 27 6, –18 –12 12 –6, 18 9, –12 –3 3 –9, 12 2 There are no factors of –108 with a sum of 0, so 4t − 27 is prime. 3 2 82. x − 3x − 9x + 27 SOLUTION: 83. POPULATION The country of Latvia has been experiencing a 1.1% annual decrease in population. In 2009, its population was 2,261,294. If the trend continues, predict Latvia’s population in 2019. SOLUTION: Use the equation for exponential decay, with a = 2,261,294, r = 0.011, and t =10. Latvia’s population in 2019 will be about 2,024,510 people. eSolutions Manual - Powered by Cognero There are more 84. TOMATOES than 10,000 varieties of tomatoes. One seed company produces seed packages Page for 37 200 varieties of tomatoes. For how many varieties do they not provide seeds? SOLUTION: 10-2 Simplifying Radical Expressions Latvia’s population in 2019 will be about 2,024,510 people. 84. TOMATOES There are more than 10,000 varieties of tomatoes. One seed company produces seed packages for 200 varieties of tomatoes. For how many varieties do they not provide seeds? SOLUTION: Let t be the number of varieties of tomatoes for which the seed company does not produce. Since the seed company produces packages for 200 varieties, the total number of varieties can be expressed as: t + 200. Since there are more than 10,000 varieties of tomatoes: t + 200 > 10,000. Solving, we get: t > 10,000 - 200 t > 9,800 Write the prime factorization of each number. 85. 24 SOLUTION: 86. 88 SOLUTION: 87. 180 SOLUTION: 88. 31 SOLUTION: The number 31 is prime. So, the prime factorization of 31 is 31. 89. 60 SOLUTION: 90. 90 SOLUTION: eSolutions Manual - Powered by Cognero Page 38