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Radical Expressions and Triangles About Radical and Chapter 12 Rational Functions RadicalRadical andand RationalRational Nonlinear functions FunctionsFunctions such as radical and rational functions can be used to model real-world situations such as the speed of a roller coaster. In this Chapter 11 unit, you will learn Radical Expressions and Triangles about radical and Chapter 12 rational functions. Chapter 12 Rational Expressions and Equations 582 Unit 4 Radical and Rational Functions Building the Best Roller Coaster USA TODAY Snapshots® A day in the park Each year, amusement park owners compete to earn What the typical family of four part of the billions of dollars Americans spend pays to visit a park1: at amusement parks. Often the parks draw customers with new taller and faster roller coasters. In this $163 project, you will explore how radical and rational $180 functions are related to buying and building a new $116 roller coaster. $120 Log on to www.algebra1.com/webquest. $60 ’01 Begin your WebQuest by reading the Task. ’99 ’97 0 ’95 Then continue working ’93 Lesson 11-1 12-2 1 — Admission for two adults and two children, parking for one car and purchase of two hot dogs, two hamburgers, four orders of fries, four small on your WebQuest as Page 590 652 soft drinks and two children’s T-shirts. you study Unit 4. Source: Amusement Business By Marcy E. Mullins, USA TODAY Unit 4 Radical and Rational Functions 583 Radical Expressions and Triangles • Lessons 11-1 and 11-2 Simplify and perform Key Vocabulary operations with radical expressions. • radical expression (p. 586) • Lesson 11-3 Solve radical equations. • radical equation (p. 598) • Lessons 11-4 and 11-5 Use the Pythagorean • Pythagorean Theorem (p. 605) Theorem and Distance Formula. • Distance Formula (p. 611) • Lessons 11-6 and 11-7 Use similar triangles and • trigonometric ratios (p. 623) trigonometric ratios. Physics problems are among the many applications of radical equations. Formulas that contain the value for the acceleration due to gravity, such as free-fall times, escape velocities, and the speeds of roller coasters, can all be written as radical equations. You will learn how to calculate the time it takes for a skydiver to fall a given distance in Lesson 11-3. 584 Chapter 11 Radical Expressions and Triangles Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 11. For Lessons 11-1 and 11-4 Find Square Roots Find each square root. If necessary, round to the nearest hundredth. (For review, see Lesson 2-7.) 1. ͙25ෆ 2. ͙80ෆ 3. ͙56ෆ 4. ͙324ෆ For Lesson 11-2 Combine Like Terms Simplify each expression. (For review, see Lesson 1-6.) 5. 3a ϩ 7b Ϫ 2a 6. 14x Ϫ 6y ϩ 2y 7. (10c Ϫ 5d) ϩ (6c ϩ 5d) 8. (21m ϩ 15n) Ϫ (9n Ϫ 4m) For Lesson 11-3 Solve Quadratic Equations Solve each equation. (For review, see Lesson 9-3.) 9. x(x Ϫ 5) ϭ 0 10. x2 ϩ 10x ϩ 24 ϭ 0 11. x2 Ϫ 6x Ϫ 27 ϭ 0 12. 2x2 ϩ x ϩ 1 ϭ 2 For Lesson 11-6 Proportions Use cross products to determine whether each pair of ratios forms a proportion. Write yes or no. (For review, see Lesson 3-6.) 2 8 4 16 8 12 6 3 13. ᎏᎏ, ᎏᎏ 14. ᎏᎏ, ᎏᎏ 15. ᎏᎏ, ᎏᎏ 16. ᎏᎏ, ᎏᎏ 3 12 5 25 10 16 30 15 Make this Foldable to help you organize information about radical expressions and equations. Begin with a sheet of plain 8ᎏ1ᎏ" by 11 paper. 2 " Fold in Half Fold Again Fold the Fold in half top to the lengthwise. bottom. Cut Label Open. Cut along the Label each Radical tab as shown. Expressions second fold to make Radical E two tabs. quations Reading and Writing As you read and study the chapter, write notes and examples for each lesson under each tab. Chapter 11 Radical Expressions and Triangles 585 Simplifying Radical Expressions • Simplify radical expressions using the Product Property of Square Roots. • Simplify radical expressions using the Quotient Property of Square Roots. Vocabulary are radical expressions used in space exploration? • radical expression A spacecraft leaving Earth must have a velocity • radicand of at least 11.2 kilometers per second (25,000 • rationalizing the miles per hour) to enter into orbit. This velocity denominator is called the escape velocity. The escape velocity • conjugate of an object is given by the radical expression 2GM Ί๶ᎏᎏ, where G is the gravitational constant, R M is the mass of the planet or star, and R is the radius of the planet or star. Once values are substituted for the variables, the formula can be simplified. PRODUCT PROPERTY OF SQUARE ROOTS Aradical expression is an expression that contains a square root. Aradicand , the expression under the radical sign, is in simplest form if it contains no perfect square factors other than 1. The following property can be used to simplify square roots. Product Property of Square Roots •Words For any numbers a and b, where a Ն 0 and b Ն 0, the square root of a product is equal to the product of each square root. • Symbols ͙abෆ ϭ ͙aෆ и ͙bෆ • Example ͙4ෆи 25 ϭ ͙4ෆ и ͙25ෆ The Product Property of Square Roots and prime factorization can be used to simplify radical expressions in which the radicand is not a perfect square. Example 1 Simplify Square Roots Simplify. a. ͙12ෆ ͙ෆ ϭ ͙ෆи и ෆ Study Tip 12 2 2 3 Prime factorization of 12 ϭ ͙2ෆ2 и ͙3ෆ Product Property of Square Roots Reading Math 2͙3ෆ is read two times ϭ 2͙3ෆ Simplify. the square root of 3 or two radical three. b. ͙90ෆ ͙90ෆ ϭ ͙2ෆи 3 и 3ෆи 5 Prime factorization of 90 ϭ ͙3ෆ2 и ͙2ෆи 5 Product Property of Square Roots ϭ 3͙10ෆ Simplify. 586 Chapter 11 Radical Expressions and Triangles The Product Property can also be used to multiply square roots. Study Tip Example 2 Multiply Square Roots ͙ෆ и ͙ෆ Alternative Method Find 3 15. To find ͙3ෆ и ͙15ෆ, you ͙3ෆ и ͙15ෆ ϭ ͙3ෆ и ͙3ෆ и ͙5ෆ Product Property of Square Roots could multiply first and then use the prime factorization. ϭ ͙3ෆ2 и ͙5ෆ Product Property ͙ෆ и ͙ෆ ϭ ͙ෆ 3 15 45 ϭ 3͙5ෆ Simplify. ϭ ͙3ෆ2 и ͙5ෆ ϭ 3͙5ෆ When finding the principal square root of an expression containing variables, be sure that the result is not negative. Consider the expression ͙xෆ2. It may seem that ͙xෆ2 ϭ x. Let’s look at x ϭϪ2. ͙xෆ2 x ͙(ෆϪ2)2 Ϫ2 Replace x with Ϫ2. ͙4ෆ Ϫ2 (Ϫ2)2 ϭ 4 2 Ϫ2 ͙4ෆ ϭ 2 For radical expressions where the exponent of the variable inside the radical is even and the resulting simplified exponent is odd, you must use absolute value to ensure nonnegative results. ͙xෆ2 ϭ x ͙xෆ3 ϭ x͙xෆ ͙xෆ4 ϭ x2 ͙xෆ5 ϭ x2͙xෆ ͙xෆ6 ϭ x3 Example 3 Simplify a Square Root with Variables Simplify ͙40ෆx4y5zෆ3. ͙40ෆx4y5zෆ3 ϭ ͙2ෆ3 и 5 иෆx4 и y5ෆи z3 Prime factorization ϭ ͙2ෆ2 и ͙2ෆ и ͙5ෆ и ͙xෆ4 и ͙yෆ4 и ͙yෆ и ͙zෆ2 и ͙zෆ Product Property ϭ 2 и ͙2ෆ и ͙5ෆ и x2 и y2 и ͙yෆ и z и ͙zෆ Simplify. ϭ 2 2 ͙ෆ The absolute value of z ensures 2x y z 10yz a nonnegative result. QUOTIENT PROPERTY OF SQUARE ROOTS You can divide square roots and simplify radical expressions that involve division by using the Quotient Property of Square Roots. Quotient Property of Square Roots •Words For any numbers a and b, where a Ն 0 and b Ͼ 0, the square root of a quotient is equal to the quotient of each square root. ͙ෆ ͙ෆ • Symbols Ί๶ᎏaᎏ ϭ ᎏa • Example Ί๶ᎏ4ᎏ9 ϭ ᎏ49 b ͙bෆ 4 ͙4ෆ Study Tip Look Back You can use the Quotient Property of Square Roots to derive the Quadratic To review the Quadratic Formula by solving the quadratic equation ax2 ϩ bx ϩ c ϭ 0. Formula, see Lesson 10-4. ax2 ϩ bx ϩ c ϭ 0 Original equation b c x2 ϩ ᎏᎏx ϩ ᎏᎏ ϭ 0 Divide each side by a, a 0. a a (continued on the next page) www.algebra1.com/extra_examples Lesson 11-1 Simplifying Radical Expressions 587 b c c x2 ϩ ᎏᎏx ϭϪᎏᎏ Subtract ᎏᎏ from each side. a a a b b2 c b2 b 2 b2 x2 ϩ ᎏᎏx ϩ ᎏᎏ ϭϪᎏᎏ ϩ ᎏᎏ Complete the square; ΂ᎏᎏ΃ ϭ ᎏᎏ. a 4a2 a 4a2 2a 4a2 b 2 Ϫ4ac ϩ b2 b b2 ΂x ϩ ᎏᎏ΃ ϭ ᎏᎏ Factor x2 ϩ ᎏᎏx ϩ ᎏᎏ. 2a 4a2 a 4a2 b b2 Ϫ 4ac x ϩ ᎏᎏ ϭ Ί๶ᎏᎏ๶ Take the square root of each side. 2a 4a2 b b2 Ϫ 4ac x ϩ ᎏᎏ ϭϮΊ๶ᎏᎏ๶ Remove the absolute value symbols and insert Ϯ. Study Tip 2a 4a2 Plus or Minus b ͙bෆ2 Ϫ 4acෆ x ϩ ᎏᎏ ϭϮᎏᎏ Quotient Property of Square Roots Symbol 2a ͙ෆ2 The Ϯ symbol is used with the 4a radical expression since both b ͙bෆ2 Ϫ 4acෆ square roots lead to solutions. x ϩ ᎏᎏ ϭϮᎏᎏ ͙4ෆa2 ϭ 2a 2a 2a 2 Ϫb ͙ෆb Ϫ 4acෆ b x ϭ ᎏᎏ Subtract ᎏᎏ from each side. 2a 2a Thus, we have derived the Quadratic Formula. A fraction containing radicals is in simplest form if no prime factors appear under the radical sign with an exponent greater than 1 and if no radicals are left in the denominator.
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