10-2 Simplifying Radicals Content Standard Prepares for A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Objective To simplify radicals involving products and quotients Use what you know about triangles to solve this problem. Suppose you are bringing a mirror into your living room. What is the maximum height of a square mirror that will fit through the doorway shown? Justify your reasoning. 2w MATHEMATICAL PRACTICES w A C T DYNAMIC I V I E S I T Dynamic Activity Simplifying Radicals Lesson Vocabulary radical expression rationalize the denominator In the Solve It, the maximum height of the mirror is a radical expression. A radical expression, such as 2 or!x 1 3, is an expression that contains a radical. A radical expression is simplified if the following statements are true. The radicand has no perfect-square factors other than 1. The radicand contains no fractions. No radicals appear in the denominator of a fraction. Simplified 3! 9!x!2 4 Not Simplified x 3!12 Å2 Essential Understanding You can simplify radical expressions using multiplication and division properties of square roots. Property Multiplication Property of Square Roots Algebra Example For a $ 0 and b $ 0,!ab!a?!b.!48!16? 4 You can use the Multiplication Property of Square Roots to simplify radicals by removing perfect-square factors from the radicand. Lesson 10-2 Simplifying Radicals 619
What strategy can you use to find the factor to remove? You can solve a simpler problem by first just listing the factors of the radicand. Then choose the greatest perfect square on the list. Problem 1 Removing Perfect-Square Factors What is the simplified form of!160?!160!16? 10 16 is the greatest perfect-square factor of 160.!16?!10 Use the Multiplication Property of Square Roots. 4!10 Simplify!16. 1. What is the simplified form of 2? Sometimes you can simplify radical expressions that contain variables. A variable with an even exponent is a perfect square. A variable with an odd exponent is the product of a perfect square and the variable. For example, n 3 n 2? n, so "n 3 "n 2? n. In this lesson, assume that all variables in radicands represent nonnegative numbers. Problem 2 Removing Variable Factors How is this problem similar to Problem 1? In both problems, you need to remove a perfect-square factor from the radicand. In this problem, however, the factor you remove contains a variable. Multiple Choice What is the simplified form of "4n 7? n 3!4n 9n 6!6n 3n 3!6n 3n!27n "4n 7 "9n 6? 6n 9n 6, or (3n 3 ) 2, is a perfect-square factor of 4n 7. "9n 6?!6n Use the Multiplication Property of Square Roots. 3n 3!6n Simplify "9n 6. The correct answer is C. 2. What is the simplified form of 2m"80m 9? You can use the Multiplication Property of Square Roots to write!a?!b!ab. Problem 3 Multiplying Two Radical Expressions What property allows you to multiply the whole numbers first? The Commutative Property of Multiplication allows you to change the order of the factors. What is the simplified form of 2t? 3"14t 2? 2t? 3 "14t 2 6 "7t? 14t 2 Multiply the whole numbers and use the Multiplication Property of Square Roots. 6"98t 3 Simplify under the radical symbol. 6"49t 2? 2t 49t 2, or (7t) 2, is a perfect-square factor of 98t 3. 6"49t 2?!2t Use the Multiplication Property of Square Roots. 6? 7t!2t Simplify "49t 2. 42t!2t Simplify. 620 Chapter 10 Radical Expressions and Equations
3. What is the simplified form of each expression in parts (a) (c)? a. 3!6?!18 b.!2a? "9a 3 c. 7!x? 3 "20x d. Reasoning In Problem 3, can you simplify the given product by first simplifying "14t 2? Explain. Problem 4 Writing a Radical Expression Art A rectangular door in a museum is three times as tall as it is wide. What is a simplified expression for the maximum length of a painting that fits through the door? w The door is w units wide and 3w units high. The diagonal length d of the doorway 3w How is this like problems you have done before? The width and height of the door are two legs of a right triangle. This is like finding the hypotenuse of a right triangle using the Pythagorean Theorem. Use the Pythagorean Theorem. d 2 w 2 1 (3w) 2 Pythagorean Theorem d 2 w 2 1 9w 2 Simplify (3w) 2. d 2 10w 2 Combine like terms. d "10w 2 Find the principal square root of each side. d "w 2?!10 Multiplication Property of Square Roots d w!10 Simplify "w 2. An expression for the maximum length of the painting is w!10, or about 3.16w. 4. A door s height is four times its width w. What is the maximum length of a painting that fits through the door? You can simplify some radical expressions using the following property. Algebra Property Division Property of Square Roots Example a For a $ 0 and b. 0, Å b!a!b. 36 Å49 6!49 6 7 When a radicand has a denominator that is a perfect square, it is easier to apply the Division Property of Square Roots first and then simplify the numerator and denominator of the result. When the denominator of a radicand is not a perfect square, it may be easier to simplify the fraction first. Lesson 10-2 Simplifying Radicals 621
Which method should you use? If the denominator is a perfect square, apply the Division Property of Square Roots first. If not, simplify the fraction first. Problem Simplifying Fractions Within Radicals What is the simplified form of each radical expression? A Å 64 49 64 Å49!64!49 B Å 8x 3 0x 8 7 Use the Division Property of Square Roots. Simplify!64 and!49. 8x 3 Å0x 4x 2 Å 2 "4x2!2!4? "x2!2 2x Divide the numerator and denominator by 2x. Use the Division Property of Square Roots. Use the Multiplication Property of Square Roots. Simplify!4, "x 2, and!2.. What is the simplified form of each radical expression? a. Å 144 9 b. Å 36a 4a 3 c. 2y 3 Å z 2 When a radicand in a denominator is not a perfect square, you may need to rationalize the denominator to remove the radical. To do this, multiply the numerator and denominator by the same radical expression. Choose an expression that makes the radicand in the denominator a perfect square. It may be helpful to start by simplifying the original radical in the denominator. Does multiplying an expression by Á 7 Á 7 change its value? No. The fraction Á 7 Á 7 is equal to 1. Multiplying an expression by 1 won t change its value. Problem 6 Rationalizing Denominators What is the simplified form of each expression? A B?!21!49!21 7 Multiply by. Multiply by!2n!2n.!8n!8n 2!2n 2!2n?!2n!2n!14n 2"4n 2!14n 4n 6. What is the simplified form of each radical expression? a.!2 b.!!18m c. Å 7s 3 622 Chapter 10 Radical Expressions and Equations
Lesson Check Do you know HOW? 1.!98 2. Í 16b 3. 3!m? 4 Å 1 m 3 4. Å 1x x 3.! 6.!6!2n Do you UNDERSTAND? 7. Vocabulary Is the radical expression in simplified form? Explain. a. 1 3 b. 7 Å 6 11 MATHEMATICAL PRACTICES c. 2!17 3 8. Compare and Contrast Simplify two different!12 ways. Which way do you prefer? Explain. 9. Writing Explain how you can tell whether a radical expression is in simplified form. Practice and Problem-Solving Exercises MATHEMATICAL PRACTICES A Practice See Problems 1 and 2. 10.!22 11.!99 12.!128 13. 2!60 14. 24!117 1. 00 16. "192s 2 17. "0t 18. 3 "18a 2 19. 221 "27x 9 20. 3 "10b 8 21. 22 "243y 3 Simplify each product. 22.!8? 2 23. 1 3!6?!24 24. 4!10? 2!90 2.!6? 1 6!216 26. 2!21? (23!42) 27.!18n? "98n3 28. 3!c? 7 "1c 2 29.!2y? "128y 30. 26 "1s 3? 2 31. 29 "28a 2? 1 3!63a 32. 10 "12x 3? 2 "6x 3 33. 2 1 3 "18c? Q26 "8c 9 R See Problem 3. STEM 34. Construction Students are building rectangular wooden frames for the set of a school play. The height of a frame is 6 times the width w. Each frame has a brace that connects two opposite corners of the frame. What is a simplified expression for the length of a brace? 3. Park A park is shaped like a rectangle with a length times its width w. What is a simplified expression for the distance between opposite corners of the park? See Problem 4. See Problems and 6. 36. Å 16 2 40. 2 Å 162t 3 2t 6 37. 7 Å32 41. 11 Å 49a 4a 3 42. 100 38. 24 Å729 1!11 39. Å 3x 3 64x 2 43.!!8x 44. 3!6!1 4. 22!11 46. 2!24 "48t 4 47. 8s "28s 3 Lesson 10-2 Simplifying Radicals 623
B Apply 48. Look for a Pattern From a viewing height of h feet, the approximate distance d to the horizon, in miles, is 3h given by the equation d Å 2. a. To the nearest mile, what is the h distance to the horizon from a height of 10 ft? 22 ft? 300 ft? b. How does the distance to the horizon increase as the height increases? 49. Think About a Plan A square picture on the front page of a newspaper occupies an area of 24 in. 2. What is the length of each side of the picture? Write your answer as a radical in simplified form. How can you find the side length of a square if you know the area? What property can you use to write your answer in simplified form? Explain why each radical expression is or is not in simplified form. 0. 13x 1. 3 2. 24! 3. 0!4 4. Error Analysis A student simplified the radical expression at the right. What mistake did the student make? What is the correct answer?. Reasoning You can simplify radical expressions with negative exponents by first rewriting the expressions using positive exponents. What are the simplified forms of the following radical expressions? x 2 = x a. "f 23 b. "x23!x 6. Sports The bases in a softball diamond are located at the corners of a 3600-ft 2 square. How far is a throw from second base to home plate? c. "a22 d. "(2m)23 "10a 21 m 21 7. Suppose a and b are positive integers. a. Verify that if a 18 and b 10, then!a?!b 6!. b. Open-Ended Find two other pairs of positive integers a and b such that!a?!b 6!. Third base Second base Home plate First base 8.!12? 9.!26? 2 60. 2!64 62.!180 63. "x2 64. 23!2 "y 3!6 66. "20a 2 b 3 67. "a 3 b c 3 68. Å 3m 16m 2 Solve each equation. Leave your answer in simplified radical form. 61. 22 "a 3 6.!8?!10 69. 16a "6a 3 70. x 2 1 6x 2 9 0 71. n 2 2 2n 1 1 72. 3y 2 2 4y 2 2 0 73. Open-Ended What are three numbers whose square roots can be written in the form a for some integer value of a? 624 Chapter 10 Radical Expressions and Equations
C Challenge 74.!24?!2x? x 7. 2b(!b) 2 76. "4a 7?!20a 77. Geometry The equation r Å A p gives the radius r of a circle with area A. What is the radius of a circle with the given area? Write your answer as a simplified radical and as a decimal rounded to the nearest hundredth. a. 0 ft 2 b. 32 in. 2 c. 10 m 2 78. For a linear equation in standard form Ax 1 By C, where A 2 0 and B 2 0, the distance d between the x- and y-intercepts is given by d Å Q C AR 2 1 Q C BR 2. What is the distance between the x- and y-intercepts of the graph of 4x 2 3y 2? Standardized Test Prep SAT/ACT Short Response 79. What is the simplified form of "12y? 2"3y 4y 4 y 2y 2 y 3y 3 80. In the proportion 3 b 7 8 2 b, what is the value of b? 6 21 8 12 12 81. The area of the triangle at the right is 24 in. 2. What is the height of the triangle? 1.8 in. 7 in. 3 in. 16 in. 82. An architect is sketching a line on a coordinate grid showing the location of a pipe. The line has an x-intercept of 22 and a y-intercept of 3. What is an equation of the architect s line? 2x 2 x 4 Mixed Review Determine whether the given lengths can be side lengths of a right triangle. See Lesson 10-1. 83. 7, 24, 2 84. 1, 4 3, 3 8., 13, 14 Factor each expression. See Lesson 8-7. 86. 64y 2 2 9 87. a 2 2 81 88. 2 2 16b 2 Get Ready! To prepare for Lesson 10-3, do Exercises 89 91. Simplify each product. See Lesson 8-3. 89. (3a 2 4)(2a 1 1) 90. (2m 2 3n)(4n 2 2m) 91. ( 1 2x)(2x 1 3) Lesson 10-2 Simplifying Radicals 62