-. Plan Lesson Preview Check Skills You ll Need Operations With Radical Epressions Lesson -: Eamples,, 7 Eercises, Etra Practice, p. 7 Lesson Preview What You ll Learn - To simplify sums and differences To simplify products and quotients... And Why To find the width of a painting, as in Eample Operations With Radical Epressions Check Skills You ll Need (For help, go to Lesson -.) Simplify each radical epression..! ".!00 0".! ". " Rationalize each denominator. "! "! "0! "0.. 7.!!8! New Vocabulary like radicals unlike radicals conjugates Lesson Resources Teaching Resources Practice, Reteaching, Enrichment Reaching All Students Practice Workbook - Spanish Practice Workbook - Basic Algebra Planning Guide - Presentation Assistant Plus! Transparencies Check Skills You ll Need - Additional Eamples - Student Edition Answers - Lesson Quiz - PH Presentation Pro CD - Computer Test Generator CD Technology Resource Pro CD-ROM Computer Test Generator CD Prentice Hall Presentation Pro CD www.phschool.com Student Site Teacher Web Code: aek-00 Self-grading Lesson Quiz Teacher Center Lesson Planner Resources Plus Part Simplifying Sums and Differences Check Understanding Need Help? Multiplication Property of Square Roots:!ab =!a?!b and!a?!b =!ab Check Understanding For radical epressions, like radicals have the same radicand. Unlike radicals do not have the same radicand. For eample,!7 and!7 are like radicals, but! and! are unlike radicals. To simplify sums and differences, you use the Distributive Property to combine like radicals. Simplify! +!. Combining Like Radicals! +! =! +! Both terms contain!. = ( + )! Use the Distributive Property to combine like radicals. =! Simplify. Simplify each epression. a.! -! 7 " b.!0 -!0 "0 You may need to simplify a radical epression to determine if you have like radicals. Simplify 7! -!. Simplifying to Combine Like Radicals Interactive lesson includes instant self-check, tutorials, and activities. 7! -! = 7! -!? is a perfect square and a factor of. = 7! -!?! Use the Multiplication Property of Square Roots. = 7! -! Simplify!. = (7 )! Use the Distributive Property to combine like radicals. =! Simplify. Simplify each epression. a.!0 +! 8 " b.! -!7 " 00 00 Chapter Radical Epressions and Equations Ongoing Assessment and Intervention Before the Lesson Diagnose prerequisite skills using: Check Skills You ll Need During the Lesson Monitor progress using: Check Understanding Additional Eamples Standardized Test Prep After the Lesson Assess knowledge using: Lesson Quiz Computer Test Generator CD
Part Simplifying Products and Quotients. Teach When simplifying a radical epression like! A! 7B, use the Distributive Property to multiply! times A! 7B. Simplify!A! 7B. Using the Distributive Property!A! 7B =!8 + 7! Use the Distributive Property. =!9?! + 7! Use the Multiplication Property of Square Roots. =! + 7! Simplify. Math Background Unlike radicals, such as! and!, are analogous to different variables, such as and y. They cannot be combined by adding or subtracting. Teaching Notes Check Understanding Check Understanding Need Help? Remember that the difference of two squares can be factored as (a + b)(a - b). Simplify each radical epression. a.!a!0b b.!a! B c. If both radical epressions have two terms, you can multiply the same way you find the product of two binomials, by using FOIL. Simplifying Using FOIL Simplify A! -!BA! +!B. A! -!BA! +!B =! +!7 -!7 -! Use FOIL. = -!7 - () Combine like radicals and simplify! and!. = -!? - 0 is a perfect square factor of 7. = -!?! - 0 Use the Multiplication Property of Square Roots. = -! - 0 Simplify!. =- -! Simplify. Simplify each radical epression. a. A! +!BA! -!B b. A!7 + B!aA!a B " ± " " " a ± "a " ± 8"7 Conjugates are the sum and the difference of the same two terms. The radical epressions! +! and! -! are conjugates. The product of two conjugates results in a difference of two squares. A!!BA!!B = A!B - A!B = - = Notice that the product of these conjugates has no radical. You recall that a simplified radical epression has no radical in the denominator. When a denominator contains a sum or a difference including radical epressions, you can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. For eample, to simplify a radical epression like, you multiply by " ". " " " " Lesson - Operations With Radical Epressions 0 Technology Tip Suggest that students use a calculator to verify that the combining of like terms is correct. Find the value of the original equation and compare it to the value of the answer. Additional Eamples Simplify! +!.! Simplify 8! -!.! Teaching Notes Inclusion Students with some kinds of vision problems will have difficulty distinguishing between the terms, especially when using FOIL, due to the distraction of so many radical signs. Suggest that they add etra space between the epressions and around each operation sign, and use arcs to connect each pair of terms to be multiplied. Additional Eamples Simplify! (!8 + 9).!0 ± 9! Simplify (! -!)(! +!). 7! Reaching All Students Below Level To emphasize to students that only like radicals can be combined show them!9 +!!. Rather!9 +! = + = 7. Advanced Learners Have students simplify. " "y English Learners See note on page 0. Inclusion See note on page 0. 0
Additional Eamples 8 Simplify.!7 ±! Á 7 Á The ratio length : width of a painting is approimately equal to the golden ratio ( +!). The length of the painting is in. Find the eact width of the painting in simplest radical form. Then find the approimate width to the nearest inch. (Á ) in.; in. English Learners The letters j and g are pronounced differently in different languages. Help students pronounce the word conjugate. Eplain that conjugate means to join together in pairs. Connection to Art Encourage students to research how the golden ratio was used in Renaissance architecture. Invite interested students to make a presentation for the class describing the golden ratio and showing pictures of buildings incorporating the golden ratio proportions. Closure Ask students to summarize how to add and subtract radicals. You remove perfect squares that are factors of the radicands, and write their square roots outside the radical signs and combine like terms. "7 "0 " 8 " 8 " " Check Understanding Simplify.!!!! Rationalizing a Denominator Using Conjugates =!!? Multiply the numerator and the denominator by the conjugate of the denominator. A!!B = Multiply in the denominator. A!! B = Simplify the denominator. = A!!B Divide and by the common factor. =!! Simplify the epression. Simplify each epression. See left. a. b. c.!7!!0!8 You can solve a ratio involving radical epressions. Real-World Problem Solving Art The ratio length : width of this painting by Mondrian is approimately equal to the golden ratio A +!B i. The length of the painting is 8 inches. Find the width of the painting in simplest radical form. Then find the approimate width to the nearest inch. Define 8 = length of painting = width of painting Relate A! B i = length i width! Write = 8 A! B = Cross multiply. A! B = Divide both sides by A! B. A! B A! B Multiply the numerator and the = A! B? A!B A!B denominator by the conjugate of the denominator. A!B = Multiply in the denominator. A!B = Simplify the denominator. 8A!B = Divide and by the common factor. Use a calculator. = 0.00709 < 0!!!!!! The eact width of the painting is painting is 0 inches. 8A!B inches. The approimate width of the Check Understanding Another painting has a length i width ratio approimately equal to the golden ratio A +!B i. Find the length of a painting if the width is inches. in. 0 Chapter Radical Epressions and Equations pages 0 0 Eercises. 9 ". 8 0"0. "7. ± "0. ± 9" 7. " 0
EXERCISES Practice and Problem Solving A B Practice by Eample Eample (page 00) Eample (page 00) Eample (page 0) Eample (page 0) Eample (page 0) Eample (page 0) Apply Your Skills Simplify each epression..! + 8! ".!0 +!0 8"0.! -! " Tell whether each pair of epressions can be simplified to like radicals. 7.!,! yes 8.!,!7 yes 9.!,!0 no Simplify each epression. 0.!8 +! ".! - 7! ".!8 +! ".! -! ".!7 -!8 "7.!0 +!0 8"0.!A!8 - B " 7.!A!7 + B 9 ± " 8.!A! - B " 9.!A! + B " ± " 0.!A +!B " ±.!A! - B ". A! +!BA! -!B. A! -!BA! -!B. - 7. See margin p. 0. A!7 B. A!0 +!B. A! + BA! + B 7. A -!BA9 +!B 8. 8 "7 ± " 9.!7!!8! " "0 " 0. 8 " ".!!8!0!. 0 " ". 9 8 " ± 9"!!!! Find an eact solution for each equation. Find the approimate solution to the nearest tenth.. See margin.!. =.!! =. =!!!! 7. The ratio of the length to the width of a painting is A! B i. The length is ft. What is the width? 7. ft Simplify each epression. 9. See margin. For more practice, see Etra Practice..!7 -!7 "7.! -! ".! -! 8" 8.!0 +!90 "0 9.!A +!B 0.! +!7 -!. A!!0!! B.. A!7 +!8BA!7 +!8B!!!!.!A! +!8B.!0-7!8.!9!. Practice Assignment Guide Objective A Objective A C B B Core, 8, Core 7, Etension 7 Standardized Test Prep 7 7 Mied Review 7 9 Error Prevention Eercises 0 Students may think a radical cannot be simplified because they choose factors that do not contain a perfect square. For eample, for!8 students might choose and instead of 9 and. Suggest to them that they factor the radicand completely and look for pairs of factors. Alternative Method Eercises 0 7 Help students see that when multiplying the square roots of identical radicands, the product is just the radicand. This method is quicker than multiplying the radicands, and then finding the square root of the product. Enrichment - Reteaching - Practice - Name Class Date Practice - Simplify each epression.. "7 + "7. 0" - ". Operations with Radical Epressions "Q "R " " ". +. + 7". "Q" "R 7. Chemistry The ratio of the rates of diffusion of two gases is given by the r formula!m = r, where m and m are the masses of the molecules of the!m r gases. Find if m = units and m = 0 units. "0 r Lesson - Operations With Radical Epressions 0. 0( " ± );. 0. ".. ;.. 8 ± ". " " "0 ". "; 0.. 8. " ± " ± " ± 9. " ± ". ± " 7. "8 + " 8. " - 8" 9. "Q" "R 0. "8 - "0. " + "8. " ". Q8" 7R. 8Q" "R. 7" - ". "Q7 "R 7. 8Q "R 8. " + "7 9. 9 " + " 0. 8 " + 0". "0Q "R. 9" - "0. 0" - 7". " - ". "7 + "8. 8" - " 7. "0 + "0 8. "Q" "R 9. "9 - "9 0. 0" - ". 8" - "7. "Q" "R. 7 " + ". "9 + "9. "9 - "9. "8Q" 7R 7. 8. 9. " " "7 " " 0. Q".. 7 R Q" "R " "7 ".. " ". " " " " Solve each eercise by using the golden ratio Q "R :.. The ratio of the height ; width of a window is equal to the golden ratio. The width of the door is in. Find the height of the door. Epress your answer in simplest radical form and in inches. 7. The ratio of the length ; width of a flower garden is equal to the golden ratio. The width of the garden is ft. Find the length of the garden. Epress your answer is simplest radical form and in feet. 8. The ratio of the width ; height of the front side of a building is equal to the golden ratio. The height of the building is 0 ft. Find the width of the building. Epress your answer in simplest radical form and in feet. Lesson - Practice Algebra Chapter Pearson Education, Inc. All rights reserved. 0
Connection to History Eercise The greatest number of kites flown on a single line is,8. Sadao Harada and a team of assistants achieved this feat in Kagoshima, Japan, in October 990. Math Tip Eercises Remind students that for some kinds of problems percents must be rounded up, disregarding the usual rounding rules. Show them that if they round down they will not reach the target amount. 8. 8" units 9. (0 ± 0 ") units 0. "0 units. ( ± "0) units Geometry Find the eact perimeter of each figure below. 8. See left. 8. y 9. y 0.. O O 0 a. The student simplified "8 as " instead of " or ". b. " ± " a. " or.8 ft. Open-Ended Make up three sums that are less than or equal to 0. Use the square roots of,,, or 7, and the whole numbers less than 0. For eample, 8" 9"7 # 0. See margin.. Error Analysis When simplifying " "8, a student wrote " ". a. What error did the student make? See left. b. Simplify " "8 correctly.. You can make a bo kite like the one at the right in the shape of a rectangular solid. The opening at each end of the kite is a square. a. Suppose the sides of the square are ft long. How long are the diagonal struts used for bracing? See left. b. Suppose each side of the square has length s. Find the length of the diagonal struts in terms of s. Write your answer in simplest form. s" Investments For Eercises 7, the formula r Î A P gives the interest rate r that will allow principal P to grow into amount A in two years, if the interest is compounded annually. Use the formula to find the interest rate you would need to meet each goal.. Suppose you have $00 to deposit into an account. Your goal is to have $9 in that account at the end of the second year. 9.%. Suppose you have $0 to deposit into an account. Your goal is to have $700 in that account at the end of two years..8% 7. Suppose you have $00 to deposit into an account. Your goal is to have $800 in that account at the end of two years..% 8. a. Suppose n is an even number. Simplify " n. b. Suppose n is an odd number greater than. Simplify " n. a"b 9. Critical Thinking Simplify. "ab b"a b n s n " s 0 0 Chapter Radical Epressions and Equations pages 0 0 Eercises. Answers may vary. Sample: 8 " ± ", "7 ± 9 ", " ± "7
0. Find the value of the numerical epression for Professor Hinkle s age in the cartoon. about years. Assess b. No; the only values it worked for were 0 and. They are unlike radicands.. Writing Eplain why! +! cannot be simplified.. a. Copy and complete the table. a b a b a b a b 0 0 "7 9 8 " 8 0 00 8 0 9 9 "8 b. Does!a +!b always equal!a b? Eplain. See left.. Error Analysis Eplain the error in the work below.! =! =! +! = + = 9 "a b u "a ± "b Lesson Quiz - Simplify each epression..! -! 0.!0 -!!.!(! +!) ±!. (! -!)(! +!) ±!7. 8! 8!7 Á Á 7 Alternative Assessment Divide the class into small groups. Tell students they will be team teachers who will teach a class what they should learn in this lesson. Have the group design a problem for each of Eamples and present to the class the problems and how to solve them. Allow groups to use the board. C Challenge Simplify each epression. 9"!8 "7.!8 +. +. +!!7 Å 8"!7!8!7 7. 8.!88 +!0 -!98 0"! Å 70. " " " ± 9. A! +!BA! +!8 +!B 70 70. 7. Find the length of each hypotenuse. Write your answers in simplified radical form. a. b. 0!!0 "!0! 0 0 " 0 " c. If the length of the legs of a right triangle are!p +!q and!p -!q, write an epression for the length of the hypotenuse. "(p q) Standardized Standardized Test Prep Test Prep Multiple Choice 7. Simplify!7 +!7. B A.! B.! C.!0 D.!0 Lesson - Operations With Radical Epressions 0 0
Standardized Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. 8 Test-Taking Strategies with Transparencies Eercise 7 Remind students to use FOIL. Encourage them to draw curved arrows from each term in the first binomial to each term in the second binomial to help in multiplying the terms together. Take It to the NET Online lesson quiz at www.phschool.com Web Code: aea-0 Short Response Etended Response Mied Review Lesson - Lesson 0- Lesson 9-7. Which radical epression is NOT equal to "? I F.!8 +!8 G.!98 -!8 H.! +! I.!8 +! 7. Simplify A! -! B A! +! B. Show your work. See back of book. 7. Eplain the steps needed to simplify. See back of book.!7! Find the distance between the points in each pair. If necessary, round to the nearest tenth. 7. (, ), (8, ) 9. units 77. (-, 7), (, 0).7 units 78. (-, ), (0, -). units Find the midpoint of each segment with the given endpoints. 79. A(, -) and B(, ) (, ) 80. H(-, ) and K(, 7) (,.) Solve each equation by factoring. 8. t - t = 0 0, 7 8. p, 9-7p - 8 = 0 8. k 9, + k + 7 = 0 8. y - y =, 8. m + 0 =-7m 8. a =-7a -, Find each product. 87 89. See margin., 87. (b + )(b + ) 88. (p + 7)(p + 7) 89. (g - 7)(g + 7) 90. ( + )( - ) 9. k 9R Q k 9R 9. (d -.)(d -.) 9 9k 8 d.d ±. Algebra at Work Auto Mechanic Auto mechanics work to see that car engines get the most out of every gallon of gasoline. Formulas used by mechanics often involve radicals. For eample, a car gets its power when gas and air in each cylinder are compressed and ignited by a spark plug. An engine s efficiency e is given c "c by the formula e = c, where c is the compression ratio. Because of the compleity of such formulas and of modern highperformance engines, today s auto mechanic must be a highly trained and educated professional who understands algebra, graph reading, and the operation of computerized equipment. Take It to the NET For more information about a career as an auto mechanic, go to www.phschool.com. Web Code: aeb-0 0 Chapter Radical Epressions and Equations pages 0 0 Eercises 87. b ± b ± 88. p ± 8p ± 9 89. g 9 0