Find each missing factor = 5 2 (j) 6. New Vocabulary rationalize the denominator EXAMPLE. 1 Multiply. Simplify if possible. b. " 3 25?

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1 7-. Plan 7- Multipling and Dividing Radical Expressions Objectives To multipl radical expressions To divide radical expressions Examples Multipling Radicals Simplifing Radical Expressions Multipling Radical Expressions Dividing Radicals Rationalizing the Denominator Real-World Connection Math Background The properties for multipling and dividing radicals, presented in this lesson, both actuall hold under less restrictive conditions n n on a and b, but more restrictive conditions on n. Students can investigate these conditions using their knowledge of imaginar numbers from Chapter. Both properties aid in simplifing radical expressions. To simplif radical expressions that involve fractions, write the expressions so that no denominator contains a radical and no radicand contains a fraction. More Math Background: p. C Lesson Planning and Resources See p. E for a list of the resources that support this lesson. 7 Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Finding Factors Algebra Review: Page 8 What You ll Learn To multipl radical expressions To divide radical expressions...an Wh To transform a famous formula, as in Example Multipling Radical Expressions 7 Chapter 7 Radical Functions and Rational Exponents Check Skills You ll Need GO for Help Find each missing factor.. 0 (j). (j) (). 8 (j). x (j) (x) x. a b (j) (b) ab. 7a 7 b 8 (j) (a) a b New Vocabular rationalize the denominator To multipl radicals consider the following.!?!9? and!? 9!. So!?!9? 9. 8? 7 -? - and 8? 7 -. So 8? 7 8? 7. In general, the product of the principal nth roots of two numbers equals the principal nth root of their product. Ke Concepts Propert Multipling Radical Expressions If n a and n b are real numbers, then n a? n b n a Multipling Radicals Multipl. Simplif if possible.!?!8!?!8!? 8!? Special Needs L Show students how a fraction can be rewritten in man equivalent forms using multiplication b.!? A!! B!? A B 0 Show how a similar procedure can be applied to radical fractions to rationalize the denominator learning stle: visual? # () - page 8 c.!?!8 The propert for multipling radicals does not appl.! is not a real number. Multipl. Simplif if possible. not possible?? 9 c.? Below Level L Have students write out the properties for multipling and dividing radical expressions using several different numerical examples. learning stle: verbal

2 is considered to be a simplified form of. You can use the propert for multipling radical expressions to simplif some radical expressions.. Teach Simplifing Radical Expressions Guided Instruction Simplif each expression. Assume that all variables Then absolute value smbols are never needed in the simplified expression. 7x 7x?? x? x Factor into perfect squares. x?!x n a? n b n ab xx Simplif. 80n 80n? 0? n? n Factor into perfect cubes. n? 0n n n a? b n ab n 0n Simplif. Simplif 0x and 8x. Assume that x is positive. Simplif the products of radicals as much as possible. x ; x8x Multipling Radical Expressions Multipl and simplif x? x. Assume that all variables x? x x? x n a? n b n ab x ( )? 0x Factor into perfect cubes. x ( )? 0x n a? n b n ab Math Tip Point out that to simplif an nth root, ou need to look for nth powers that are factors of the radicand. For example, if n, then look for factors of the radicand that are perfect squares. Additional Examples Multipl. Simplif if possible.!?!? c.!?! The Propert for Multipling Radicals does not appl. Simplif each expression. Assume all variables 0x x!x n 8 n n Multipl and simplif x 8? x. Assume all variables x x x 0x Simplif. Multipl and simplif 7x? x. Assume that all variables x Dividing Radical Expressions To divide radicals, consider the following.! and!. So.! Ä Q R! In general, the quotient of the principal nth roots of two numbers equals the principal nth root of their quotient. Ke Concepts Propert Dividing Radical Expressions If and are real numbers and b 0, then n a n n a n b a b. n b Lesson 7- Multipling and Dividing Radical Expressions 7 Advanced Learners L Point out to students that rationalizing a denominator is a necessar algebraic skill in some higher mathematics. learning stle: verbal English Language Learners ELL Review for students the different processes the are using: multiplication, division, and simplification (finding roots). Clarif that rationalizing the denominator is a simplification process that involves rewriting the radical in an equivalent form. learning stle: verbal 7

3 Guided Instruction Alternative Method The expression in part b is simplified using Method II. However, the expression can also be simplified with Method I b rewriting the expression as the square root of a fraction. Additional Examples Divide and simplif. Assume all variables 8 9x 8 x x x Rationalize the denominator of each expression. Assume all variables x xx x 0 c. The distance d in meters that an object will fall in t seconds is given b d.9t. Express t in terms of d and rationalize the 0d denominator. t 7 Resources Dail Notetaking Guide 7- L Dail Notetaking Guide 7- Adapted Instruction L Closure Dividing Radicals Divide and simplif. Assume that all variables x x x? x? x Ä x Divide and simplif. Assume that all variables! x c. 0x xx!7 x x x x To rationalize the denominator of an expression, rewrite it so there are no radicals in an denominator and no denominators in an radical. Rationalizing the denominator of a numerical expression makes it easier to calculate its decimal approximation. For example, and it is easier to!! divide b than b!. Rationalizing the Denominator Rationalize the denominator of each expression. Assume that all variables!! Method 8 - x x x x!!?? Ä!! Method!!?!!!?! x!x x!x! Rewrite as a square root of a fraction. Then make the denominator a perfect square. Multipl the numerator and denominator b! so the denominator becomes a whole number. x?!x x x!!x?!x x x x! Ask students to describe how to multipl and divide two nth roots, both of which are real numbers. To multipl the nth roots, take the nth root of the product of the radicands. To divide the nth roots, take the nth root of the quotient of the radicands. c. x x? x 8x Äx? x Ä x 8x x Rewrite the fraction so the denominator is a perfect cube. Rationalize the denominator of each expression. Assume that the variables 7 x x c. 8x 0x x x 7 Chapter 7 Radical Functions and Rational Exponents 7

4 Exercises EXERCISES Example (page 7) Example (page 7) Example (page 7) Example (page 7) Example (page 7) Real-World Connection Einstein s famous formula E mc relates energ E, mass m, and the speed of light c. Express c in terms of E and m and rationalize the denominator. E mc c E m Practice and Problem Solving A GO Practice b Example for Help c E Em!Em m Ä m m!em m d The formula a relates the acceleration a of a moving object to the distance d it t moves in the time t. Solve the formula for t and rationalize the denominator. t da a For more exercises, see Extra Skill and Word Problem Practice. 8? 0 Multipl, if possible. Then simplif..!8?!.?. 9? ?.!?!.? 7. 9? 8.? 8 not possible Simplif. Assume that all variables 9. See margin. 9. 0x 0. 8x. 0x. a a b 7. 0x. x 0a b b x Multipl and simplif. Assume that all variables x 7.? x? x 9 7x 0.!x? x 0x.? 0 0. x? x x 0x Divide and simplif. Assume that all variables!00 8x x 0x.. x.. 0! x 7x x Rationalize the denominator of each expression. Assume that all variables 0. See margin.!x x! 0x x x 0.!!8x x x. x. 0x x.. x x x. Practice Assignment Guide A B -, 7-, -9 A B -, -, 0 C Challenge -8 Test Prep 9-7 Mixed Review 7-9 Homework To check students understanding of ke skills and concepts, go over Exercises,,,, 7, 9. Error Prevention! Exercises 9 Remind students that the do not need to use absolute value signs since the variables Connection to Phsics Exercise You ma wish to point out that the distance r is the distance between the centers of the objects. GPS Enrichment Guided Problem Solving Reteaching Practice Name Class Date Practice 7- Multipling and Dividing Radical Expressions Multipl and simplif. Assume that all variables.?. 9x? 9. 0x z? z. x? 8x. x? x. 9? Rationalize the denominator of each expression. Assume that all variables 7. 9x!x x Ä!x x 0.. x. a x Ä 8 Äb c Multipl. Simplif if possible.assume that all variables.?. 8?.? 7 L L L L Example (page 77) Gm. Phsics The formula F m relates the gravitational force F between r an object of mass m and an object of mass m separated b distance r. G is a constant known as the constant of gravitation. Solve the formula for r. Rationalize the denominator. r Gm m F F Lesson 7- Multipling and Dividing Radical Expressions 77 Pearson Education, Inc. All rights reserved..? 9 7. x? x 8. x? x 7 Simplif. Assume that all variables 9. x 0. z. 8k. a. x 0 z. s 7 t. x. 7r 7. u v 8 Divide and simplif. Assume that all variables x x k x x k 7 (x) 8... a () Ä a. The volume of a sphere of radius r is V. pr Use the formula to find r in terms of V. Rationalize the denominator. Use our answer to part (a) to find the radius of a sphere with volume 00 cubic inches. Round to the nearest hundredth. 9. xx. aa. x 0. x 0. x.. x x x. 0 x0. 77

5 . Assess & Reteach c. Answers ma var. Sample: First simplif the denominator. Since , to rationalize the denominator, multipl the fraction b. This ields 78 Lesson Quiz Assume that all variables are positive.. Multipl. Simplif if possible.!?!? Simplif. 8x x!x x x 0 9. Multipl and simplif. 8x? x x!x 0x? x x x. Divide and simplif.. Rationalize the denominator of each expression. 8x x 70x 0x 7x x x x 8x Alternative Assessment Have students work in pairs. Each student should make up a problem similar to those in each of Examples. Then have students work each other s problems and check each other s work. # # 7 #. GO B 0. 0x. 0 ± 7. ±. ± Real-World Appl Your Skills nline Homework Help Visit: PHSchool.com Web Code: age-070 Connection A satellite being launched from the cargo ba of the space shuttle C Challenge 7. a c, b d 78 Chapter 7 Radical Functions and Rational Exponents. 0 cm 7. A product of two square roots can be simplified in this wa onl if the square roots are real numbers. and 8 are not.!!. Simplif b multipling the numerator and denominator b!7.!7 Simplif the expression in (a) b multipling b! instead of!7.!! c. Explain how ou would simplif. a c. See margin.!98 Simplif each expression. Rationalize all denominators. Assume that all variables 7.!?!0 0 8.? 80 x 9. x? x 7 0. x? x.!a!0 7B. (!).!(!!). x?? x. x? x? 9x x. x x x x0! 7. x 8. x x!7x x 9x x x. x x x x x 7x x x x!!!. Q R..!!8. Satellites The circular velocit v, in miles per hour, of a satellite orbiting GPS. 0 Earth is given b the formula v Î r, where r is the distance in miles from the satellite to the center of Earth. How much greater is the velocit of a satellite orbiting at an altitude of 00 mi than one orbiting at an altitude of 00 mi? (The radius of Earth is 90 mi.) mi/h greater. Geometr A rectangular shelf is!0 cm b!0 cm. Find its are 7. Error Analsis Explain the error in this simplification of radical expressions.!?!8!(8)! 7. See margin. 8. Phsics A freel falling object hit the ground in 8a seconds. It fell h feet. Use the formula h t to find h in terms of 88 a ft 9. Writing Does x x for all, some, or no values of x? Explain. See margin. 0. Open-Ended Of the equivalent expressions Î,!!, and, which do ou! prefer to use for finding a decimal approximation with a calculator? Justif our reasoning. Check students work. Simplif each expression. Rationalize all denominators. Assume that all variables. ÎÍx x. Î Íx x. ÎÍ x x. x x x x. Äx x 7. Critical Thinking When x a b is simplified, the result is, where c and d x c d are positive integers. Express a in terms of c, and b in terms of d. 8. Critical Thinking In Example ou saw that x? x simplifies to x 0x, if ou assume that all the variables Now assume that the variables represent an real numbers. What changes must be made in the answer? Explain. See margin. 9. For some values, it is eas to see that the equation is true if x 0 or x. But when x R 0, x is not a real number, although x is. 8. No changes need to be made; since the are both odd roots, there is no need for absolute value smbols.

6 Test Prep Multiple Choice 9. Which expression does NOT simplif to -0? C A. 000 B.!? 8 C.!? D.? 70. How can ou write with a rationalized denominator? x H 0x F. G. 0 H. x x x J. x x Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, p. 7 Test-Taking Strategies, p. Test-Taking Strategies with Transparencies GO Short Response Extended Response Mixed Review for Help Lesson 7- Lesson - Lesson -7 lesson quiz, PHSchool.com, Web Code: aga-070! 7. What is the simplified form of?!! A. A! B.! C.! D. 7. To rationalize the denominator of, b what number would ou multipl 9 the numerator and denominator of the fraction? G F. G. H. J Which of the following expressions is in simplest form? D A. 0x B. 8x C. D. 7. For what values of x is!x?!x a real number? Explain. 7. Rationalize the denominator of x. Explain our steps. Simplif each radical expression. Use absolute value smbols as needed. 7. a 90»a 77. 8c 8 d 9c d 78. a 8 a x 0.»x 8. 7 x x 8. x 9»x x x 0 Divide. Tell whether each divisor is a factor of the dividend. 8. ( - ) ( + ) 8. (x + 7) (x + ) 8. (a + a - a + ) (a + ) 87. (x - x - x + 0) (x - ) See margin. Complete the square. 88. x + 0x + j 89. x - 0x + j 90. x + x + j 9. x - x + j 9. x - x + j 9. x + 0.x + j x - x + j 9 9. x + 9 x + j 00 Lesson 7- Multipling and Dividing Radical Expressions 79! 7 7. See margin. 7. [] x is a real number if x L 0 and x is a real number if x K 0. So the onl value that makes x x a real number is x 0. [] answer onl OR error describing value(s) of variables 7. [] You should multipl b x because Ä x x x Ä x x x x x x x, 8x x which has a denominator without a radical. [] appropriate methods, but with one minor error [] major error, but subsequent steps consistent with that error [] correct final expression, but no work shown 8. ±, R 8, not a factor 8. x x 9, a factor 8. a 8a, R, not a factor 87. x ± x ± x, R 0, not a factor 79