CC-20 CC Objective MATHEMATICAL PRACTICES. Lesson. L Vocabulary. ANSWER CONNECT THE MATH n

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1 1 Interactive Learning Solve It! PURPOSE PROCESS 2 FACILITATE Q [to divide into two equal parts] Q a [8] Q a b [ 1 2 a] ANSWER CONNECT THE MATH n 2 Guided Instruction Problem 1 Q [3] Q [4] 7-5 CC-20 CC-9 Lesson L Vocabulary V indexi How do you find n a for any number a? You look for n equal factors of a. Rational Exponents and Radicals Objective To rewrite expressions involving radicals and rational exponents In this lesson, you will learn the relationship between radical expressions and expressions using rational exponents. Essential Understanding You can use rational exponents to represent radicals. In a radical expression, the number under the radical sign a is the radicand. The number n in the crook of the radical sign is the index. The index gives the degree of the root. For a cube root, the degree is 3. If there is no index, the degree is 2, which means square root. Recall what you know about square roots. Since , you know that You also know that Using the transitive property of equality, you can conclude that Similarly, You can simplify radical expressions by finding like factors, just as when simplifying powers with rational exponents. index radical sign n a radicand Problem 1 Finding Roots What is the simplified form of each expression? A B 4 16 Method 1 Method ( ) Content Standard N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. b c a 16 1 Common Core CC-9 Preparing to Teach BIG idea Properties ESSENTIAL UNDERSTANDINGS Math Background Mathematical Practice Attend to precision. 1 Common Core

2 In radical form, do you use the numerator or denominator as the index? You use the denominator of the rational exponent as the index of the radical. Can you write 3 27d 5 as 27d 5 3? No. Since the coefficient 27 is under the radical, the index operates on the coefficient as well as the variable. 1. What is the simplified form of each expression? a b c d You can also write expressions that have rational exponents like 2 3 in radical form (8 2 ) (8 1 3 ) 2 ( 3 8) 2 So, ( 3 8) 2. Problem 2 Problem 3 Key Concept Equivalence of Radicals and Rational Exponents If the nth root of a is a real number and m and n are positive integers, then a 1 n n a and a m n n a m ( n a) m. A What is 12a 2 3 in radical form? Converting to Radical Form 12a a 2 Rewrite a 2 3 in radical form. B What is (64a) 4 5 in radical form? (64a) (32 2a) 5 Since 32 is the 5th power of 2, write 64a as a product of 32 and 2a (2a) 5 Power of a product (2a) 5 Rewrite 32 as (2a) 4 5 Simplify (2a) 4 Simplify and write (2a) 4 5 in radical form. 2. What is each exponential expression in radical form? a. a 5 6 b. 5x 1 3 c. (54y) 2 3 Converting to Exponential Form A What is 5 b 3 in exponential form? 5 b 3 b 3 5 Rewrite using exponential form. B What is 3 27d 5 in exponential form? Simplify. 3 27d 5 (27d 5 ) (d 5 ) 1 3 3d 5 3 Rewrite the radical expression in exponential form. Power of a product Simplify. Problem 1 ERROR PREVENTION Take note VISUAL LEARNERS a w z z a w Problem 2 Q [the numerator] Q [the denominator] b 1 2 b b 1 2 b 2 1 Problem 3 ERROR PREVENTION Q 27d d 5 [The expression inside the parentheses is a product of two factors, 27 and d 5. The power of a product property requires that the root be taken for each factor.] CC-9 Rational Exponents and Radicals 2 Answers Solve It! a 82 b 8 c 2 1. a. 3 b. 2 c. d a. 6 a 5 b. 5 3 x c. 9 3 y 2 CC-9 2

3 Problem 4 Q [18 3 4] Q M [Answers will vary. Sample: 16] How can you find the approximate value of the expression? You can use 18^(3/4) to simplify the radical using a calculator. 3. Write each radical expression in exponential form. a. 3 s 2 b x 4 c. (4y) 5 d a 8 Problem 4 Using a Radical Expression STEM Biology You can estimate the metabolic rate of living organisms based on body mass using Kleiber s law. The formula R M 3 relates metabolic rate R measured in Calories per day to body mass M measured in kilograms. What is the metabolic rate of a dog with a body mass of 18 kg? R M Substitute 18 for M Use a calculator to simplify. The metabolic rate is about 641 Calories per day. 4. What is the metabolic rate of a man with a body mass of 75 kg? 3 Lesson Check Do you know HOW? Do you UNDERSTAND? Close Lesson Check Do you know HOW? Simplify each expression Write each expression using rational exponents in radical form and each radical expression in exponential form. 4. x 5. c (8d) y 3 Practice and Problem-Solving Exercises Do you UNDERSTAND? 8. Error Analysis What is the error in the problem at the right? What is the correct answer? 9. Write a rule for multiplying two radicals with the same radicand. Justify why your rule works. 10. Does ? Explain why or why not. A Practice What is the value of each expression? See Problem Q [Answers may vary.] Write each expression in radical form. 17. a (64b) x z (25x) a (98d) b (24c) 2 3 See Problem 2. 3 Common Core Additional Problems 1. a b ANSWER a. 7 b. 2. a. x 3 5 b. (8a) 2 3 ANSWER a. 5 x 3 b. 3 a 2 3. a. 7 y b. 5 32b 3 ANSWER a. y 7 b. 2b R M 3 ANSWER Answers 3. a. s 2 3 b. 12x 3 c. 32y 5 2 d Lesson Check x c 6. 3 d y y 3 (27y) y 2 3 Common Core

4 B Apply STEM Write each expression in exponential form a (2c) a (8x) c y x x b Manufacturing A company that manufactures memory chips for digital cameras uses the formula c n to determine the cost c, in dollars, of producing n chips. How much will it cost to produce 250 chips? 36. Archaeology Carbon-14 is present in all living organisms and decays at a predictable rate. To estimate the age of an organism, archaeologists measure the amount of carbon-14 left in its remains. The approximate amount of carbon-14 remaining after 5000 years can be found using the formula A A 0 (2.7) 3 5, where A0 is the initial amount of carbon-14 in the sample that is tested. How much carbon-14 is left in a sample that is 5000 years old and originally contained grams of carbon-14? Simplify each expression using the properties of exponents, and then write the expression in radical form. 37. x 3 4 x a 2 3 a cd 1 2 d x 1 3 (8x 2 ) x x x 2 3 8x 1 3 Write each expression in exponential form. Simplify when possible b 2 3 b a a b a (9x) y y 3 y 3 y 48. (2c) 4 3 c Sports The radius r of a sphere that has volume V is r 3 3V 4. The volume of a basketball is approximately in. 3. The radius of a tennis ball is about one fourth the radius of a basketball. Find the radius of the tennis ball. 50. a. Show that x 2 x by rewriting x 2 in exponential form. b. Show that 4 x 2 x by rewriting 4 x 2 in exponential form. See Problem 3. See Problem 4. 4 Practice ASSIGNMENT GUIDE Mathematical Practices Applications STEM HOMEWORK QUICK CHECK 51. Think about a Plan You want to simplify the expression 4x 3 2 3x 3. How can you write the radical expression using a rational exponent? Can you add the resulting terms? What is the result in simplest form? Can you write the result in two equivalent forms? 52. Open-Ended Write an expression using rational exponents. Then write an equivalent expression using radicals. CC-9 Rational Exponents and Radicals ; Practice and Problem-Solving Exercises a (b) x z x a d b c a c a x c y x x b x a c 6 d x x 42. 2x 43. b 2 3 b a b 5 3 a x 1 2 5y y 48. 5c a. x 2 (x 2 ) 1 2 x 21 2 x b. 4 x 2 (x 2 ) 1 x 21 x 2 x 1 2 x 51. 3x 3 3x 3 2x 3 2 3x 3 2 7x 3 2 7x 3 2 7x CC-9 4

5 Answers Practice and Problem-Solving Exercises a. b. c I 57. [2] 7s t x d c a y t 2/3 66. x y x y x 6y x 3y x 36y C Challenge Short Response 53. Inflation The formula C c(1 r) n can be used to estimate the future cost C of an item due to inflation. Here c represents the current cost of the item, r is the rate of inflation, and n is the number of years for the projection. Suppose a video game system costs $299 now. How much will the price increase in nine months with an annual inflation rate of 3.2%? 54. Cells The number of cells in a cell culture grows exponentially. The number of cells in the culture as a function of time is given by the expression N 6 5 t, where t is measured in hours and N is the initial size of the culture. a. After 2 hours, there were 144 cells in the culture. What was N? b. How many cells were in the culture after 20 minutes? c. How many cells were in the culture after 2.5 hours? Standardized Test Prep 55. Which of the following expressions is equivalent to (8x) 4 3? 16 3 x 4 3 8x x x Which of the following expressions is equivalent to 4b 5? 2b 2 5 (2b) 2 5 (4b) 5 2 4b Write the expression 9s 3 16s 3 with rational exponents. Mixed Review Simplify the following expressions (t 2 ) (2x 4 ) (a 0 ) (9d) (c 2 ) (10a 3 ) (y 4 ) (27t) Get Ready! To Prepare for Lesson 7-6, do Exercises Describe the pattern for each sequence. Then write the next three numbers in the sequence , 6, 9, , 4, 2, , 1 3, 1 4, , 7, 12, , 1, 5, , 4, 9,... See Lesson 7-3. Write each equation in standard form. See Lesson y 4x y 3 2(x 5) 68. 3y 6 7x y 5 6(x 2) 70. x 6y y 9x 4 27 See Lesson Common Core 5 Common Core