# Rules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER

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1 Rules of Exponents CHAPTER 5 Math at Work: Motorcycle Customization OUTLINE Study Strategies: Taking Math Tests 5. Basic Rules of Exponents Part A: The Product Rule and Power Rules Part B: Combining the Rules 5.2 Integer Exponents Part A: Real-Number Bases Part B: Variable Bases 5.3 The Quotient Rule Putting It All Together 5.4 Scientific Notation Group Activity empowerme: Math Test Preparation Checklist Let s take another look at how math is used in customizing motorcycles. One of Frank Esposito s coworkers at the motorcycle shop is Tanya Sorello, who is responsible for the frames of the motorcycles. Like Frank, Tanya uses mathematical equations to ensure that she does her work correctly. For example, Tanya is building a chopper frame and needs to make the supports for the axle. To do this, she has to punch -in.-diameter holes in 3 -in.-thick mild steel plates that will be welded to the frame. The press punches 8 two holes at a time. To determine how much power is needed to do this job, she uses a formula containing an exponent, P t2 dn. After substituting the numbers into the expression, she calculates that the power needed to punch these 3.78 holes is 0.07 hp. Tanya is used to working under tight deadlines. She has found ways to manage stress and perform at her best when it matters most. Tanya says she learned these skills when she was in school, during exam time. Mastering the high-pressure situation of sitting down to take a final exam has helped her succeed in critical moments throughout her career. In this chapter, we will learn about working with exponents and introduce strategies you can use when you take a math test. 307

5 Read the explanations, follow the examples, take notes, and complete the You Trys. Evaluate Exponential Expressions Recall from Chapter that exponential notation is used as a shorthand way to represent a multiplication problem. For example, can be written as 3 5. Definition An exponential expression of the form a n, where a is any real number and n is a positive integer, is equivalent to a a a a. We say that a is the base and n is the exponent. n factors of a We can also evaluate an exponential expression. EXAMPLE In-Class Example Identify the base and the exponent in each expression and evaluate. a) 3 4 b) ( 3) 4 c) 3 4 Answer: a) base: 3; exponent: 4; 8 b) base: 3; exponent: 4; 8 c) base: 3; exponent: 4; 8 In part c), you must follow the order of operations and evaluate the exponential expression before multiplying by. Identify the base and the exponent in each expression and evaluate. a) 2 4 b) ( 2) 4 c) 2 4 Solution a) is the base, 4 is the exponent. Therefore, b) ( 2) 4 2 is the base, 4 is the exponent. Therefore, ( 2) 4 ( 2) ( 2) ( 2) ( 2) 6. c) 2 4 It may be very tempting to say that the base is 2. However, there are no parentheses in this expression. Therefore, 2 is the base, and 4 is the exponent. To evaluate, The expressions ( a) n and a n are not always equivalent: ( a) n ( a) ( a) ( a) ( a) n factors of a a n a a a a n factors of a [ YOU TRY ] Identify the base and exponent in each expression and evaluate. a) 5 3 b) 8 2 c) a b SECTION 5.A Basic Rules of Exponents: The Product Rule and Power Rules 3

6 2 Use the Product Rule for Exponents Is there a rule to help us multiply exponential expressions? Let s rewrite each of the following products as a single power of the base using what we already know: factors factors 4 factors factors of 2 of 2 of 5 of 5 ) ) Let s summarize: , Do you notice a pattern? When you multiply expressions with the same base, keep the same base and add the exponents. This is called the product rule for exponents. Property Product Rule Let a be any real number and let m and n be positive integers. Then, a m a n a m n EXAMPLE 2 In-Class Example 2 Find each product. a) b) y 4 y 9 c) 4x 5 ( 0x 8 ) d) d d 7 d 4 Answer: a) 25 b) y 3 c) 40x 3 d) d 2 Find each product. a) b) x 9 x 6 c) 5c 3 7c 9 d) ( k) 8 ( k) ( k) Solution a) Since the bases are the same, add the exponents. b) x 9 x 6 x 9 6 x 5 If you do not see an exponent on the base, it is assumed that the exponent is. Look closely at part d). [ YOU TRY 2 ] c) 5c 3 7c 9 (5 7)(c 3 c 9 ) Use the associative and commutative properties. 35c 2 d) ( k) 8 ( k) ( k) ( k) 8 ( k) 20 Product rule Find each product. a) b) y 0 y 4 c) 6m 5 9m d) h 4 h 6 h 4 e) ( 3) 2 ( 3) 2 Can the product rule be applied to ? No! The bases are not the same, so we cannot add the exponents. To evaluate , we would evaluate and , then multiply: Use the Power Rule (am ) n a mn What does (2 2 ) 3 mean? We can rewrite (2 2 ) 3 first as Notice that (2 2 ) , or This leads us to the basic power rule for exponents: When you raise a power to another power, keep the base and multiply the exponents. 32 CHAPTER 5 Rules of Exponents

7 Property Basic Power Rule Let a be any real number and let m and n be positive integers. Then, (a m ) n a mn EXAMPLE 3 In-Class Example 3 Simplify using the power rule. a) (4 6 ) 3 b) (m 2 ) 5 c) (q 8 ) 7 Answer: a) 4 8 b) m 0 c) q 56 [ YOU TRY 3 ] Simplify using the power rule. a) (3 8 ) 4 b) (n 3 ) 7 c) (( f ) 4 ) 3 Solution a) (3 8 ) b) (n 3 ) 7 n 3 7 n 2 c) (( f ) 4 ) 3 ( f ) 4 3 ( f ) 2 Simplify using the power rule. a) (5 4 ) 3 b) ( j 6 ) 5 c) (( 2) 3 ) 2 This property demonstrates how you can distribute an exponent to the bases if the bases are being multiplied. 4 Use the Power Rule (ab)n a n b n We can use another power rule to simplify an expression such as (5c) 3. We can rewrite and simplify (5c) 3 as 5c 5c 5c c c c 5 3 c 3 25c 3. To raise a product to a power, raise each factor to that power. Property Power Rule for a Product Let a and b be real numbers and let n be a positive integer. Then, (ab) n a n b n Notice that (ab) n a n b n is different from (a b) n. (a b) n a n b n. We will study this in Chapter 6. EXAMPLE 4 In-Class Example 4 Simplify each expression. a) (4x) 3 b) a 4 2 ub c) ( 3g) 3 d) (2w 4 ) 5 e) 2(8mn) 2 Answer: a) 64x 3 b) c) 27g 3 d) 32w 20 e) 28m 2 n 2 6 u4 Simplify each expression. a) (9y) 2 b) a 3 4 tb Solution c) (5c 2 ) 3 d) 3(6ab) 2 a) (9y) y 2 8y 2 b) a 3 4 tb a 3 4 b t 3 64 t3 c) (5c 2 ) (c 2 ) 3 25c c 6 d) 3(6ab) 2 3[6 2 (a) 2 (b) 2 ] The 3 is not in parentheses; therefore, it will not be squared. 3(36a 2 b 2 ) 08a 2 b 2 SECTION 5.A Basic Rules of Exponents: The Product Rule and Power Rules 33

8 [ YOU TRY 4 ] Simplify. a) (k 4 ) 7 b) (2k 0 m 3 ) 6 c) ( r 2 s 8 ) 3 d) 4(3tu) 2 5 Use the Power Rule aa b b n an b n where b 0 Another power rule allows us to simplify an expression like a 2 x b 4. We can rewrite and simplify a 2 4 x b as 2 x 2 x 2 x 2 x x x x x 24 x 6. To raise a quotient to a power, 4 4 x raise both the numerator and denominator to that power. Property Power Rule for a Quotient Let a and b be real numbers and let n be a positive integer. Then, a a b b n an b n, where b 0 EXAMPLE 5 In-Class Example 5 Simplify using the power rule for quotients. a) a b b) a x 2 6 b c) a u 7 v b Answer: a) c) u7 v 7 b) x2 36 [ YOU TRY 5 ] Simplify using the power rule for quotients. a) a b b) a 5 3 x b c) a t 9 u b Solution a) a b b) a 5 x b 3 Simplify using the power rule for quotients. a) a b b) a 2 5 d b c) a u 6 v b 53 x 3 25 x 3 c) a t u b 9 t9 u 9 Let s summarize the rules of exponents we have learned in this section: Summary The Product and Power Rules of Exponents In the rules below, a and b are any real numbers, and m and n are positive integers. Rule Example Product rule a m a n a m n p 4 p p 4 p 5 Basic power rule (a m ) n a mn (c 8 ) 3 c 8 3 c 24 Power rule for a product (ab) n a n b n (3z) z 4 8z 4 Power rule for a quotient a a n b b an 4 b n, (b 0) aw 2 b w4 2 w CHAPTER 5 Rules of Exponents

9 ANSWERS TO [YOU TRY] EXERCISES ) a) base: 5; exponent: 3; b) base: 8; exponent: 2; c) base: 2 ; exponent: 3; 3 a b 8 2) a) 27 b) y 4 c) 54m 6 d) h 4 e) 8 3) a) 5 2 b) j 30 c) ) a) k 28 b) 64k 60 m 8 c) r 6 s 24 d) 36t 2 u 2 5) a) b) 32 5 d c) u6 v 6 5.A Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective : Evaluate Exponential Expressions Rewrite each expression using exponents. ) ) ) a 7 b a 7 b a 7 b a 7 b a 7 b 4 28) Is there any value of a for which ( a) 2 a 2? Support your answer with an example. Yes. If a 0, then ( 0) 2 0 and Evaluate. 29) ) ) () ) ) (0.8)(0.8)(0.8) (0.8) 3 5) ( 5)( 5)( 5)( 5)( 5)( 5)( 5) ( 5) 7 6) ( c)( c)( c)( c)( c) ( c) 5 7) ( 3y)( 3y)( 3y)( 3y)( 3y)( 3y)( 3y)( 3y) ( 3y) 8 8) a 5 4 tb a 5 4 tb a 5 4 tb a 5 4 tb 4 a 5 4 tb Identify the base and the exponent in each. 9) 6 8 base: 6; exponent: 8 0) 9 4 base: 9; exponent: 4 ) (0.05) 7 2) (0.3) 0 base: 0.05; exponent: 7 base: 0.3; exponent: 0 3) ( 8) 5 4) ( 7) 6 base: 8; exponent: 5 base: 7; exponent: 6 5) (9x) 8 6) (3k) 3 base: 9x; exponent: 8 base: 3k; exponent: 3 7) ( a) 2 8) ( 2w) 9 base: a; exponent: 2 base: 2w; exponent: 9 9) 5p 4 base: p; exponent: 4 20) 3m 5 base: m; exponent: 5 2) 3 8 y2 base: y; exponent: 2 22) 5 9 t7 base: t; exponent: 7 23) Evaluate (3 4) 2 and Are they equivalent? Why or why not? 24) Evaluate (7 3) 2 and Are they equivalent? Why or why not? 25) For any values of a and b, does (a b) 2 a 2 b 2? Why or why not? Answers may vary. 26) Does 2 4 ( 2) 4? Why or why not? 27) Are 3t 4 and (3t) 4 equivalent? Why or why not? No, 3t 4 3 t 4 ; (3t) t 4 8t 4 33) ( 2) ) ( 5) ) ) ) ) ) a 5 b ) a 3 2 b For Exercises 4 44, answer always, sometimes, or never. 4) Raising a negative base to an even exponent power will always, sometimes, or never give a negative result. never 42) If the base of an exponential expression is, the result will always, sometimes, or never be. always 43) If b is any integer value except zero, then the exponential expression ( b) 3 will always, sometimes, or never give a negative result. sometimes 44) If a is any integer value except zero, then the exponential expression a 4 will always, sometimes, or never give a positive result. never Objective 2: Use the Product Rule for Exponents Evaluate the expression using the product rule, where applicable. 45) ) ) ) ) ) ) a 4 2 b a 2 2 b 64 52) a 4 3 b a4 3 b SECTION 5.A Basic Rules of Exponents: The Product Rule and Power Rules 35

10 Simplify the expression using the product rule. Leave your answer in exponential form. 53) ) ) ) ) ( 7) 2 ( 7) 3 ( 7) 3 58) ( 3) 5 ( 3) ( 3) 6 59) b 2 b 4 ( 7) 8 b 6 60) x 4 x 3 ( 3) 2 x 7 6) k k 2 k 3 k 6 62) n 6 n 5 n 2 n 3 63) 8y 3 y 2 8y 5 64) 0c 8 c 2 c 0c 65) (9m 4 )(6m ) 54m 5 66) ( 0p 8 )( 3p) 30p 9 67) ( 6r)(7r 4 ) 42r 5 68) (8h 5 )( 5h 2 ) 40h 7 69) ( 7t 6 )(t 3 )( 4t 7 ) 28t 6 70) (3k 2 )( 4k 5 )(2k 4 ) 24k 7) a 5 3 x2 b(2x)( 2x 3 ) 72) a 7 0 y9 b( 2y 4 )(3y 2 ) 40x y5 73) a 8 2 bb( 6b8 )a 7 2 b6 b 74) (2c 3 )a 4 5 c2 b a 5 7 c6 b 8c Mixed Exercises: Objectives 3 5 Simplify the expression using one of the power rules. 75) (y 3 ) 4 y 2 76) (x 5 ) 8 x 40 77) (w ) 7 w 77 78) (a 3 ) 2 a 6 79) (3 3 ) ) (2 2 ) 2 6 8) (( 5) 3 ) 2 ( 5) 6 82) (( 4) 5 ) 3 ( 4) 5 83) a 3 b 4 85) a 6 a b 2 87) a m n b 5 8 8b 5 84) a 5 2 b ) a v 3 a 2 4 b m 5 88) a t 2 n 5 u b 25 8 v 3 64 t 2 u 2 89) (0y) 4 0,000y 4 90) (7w) 2 49w 2 9) ( 3p) 4 8p 4 92) (2m) 5 32m 5 93) ( 4ab) 3 64a 3 b 3 94) ( 2cd) 4 6c 4 d 4 95) 6(xy) 3 6x 3 y 3 96) 8(mn) 5 8m 5 n 5 97) 9(tu) 4 9t 4 u 4 98) 2(ab) 6 2a 6 b 6 Mixed Exercises: Objectives ) Find the area and perimeter of each rectangle. a) b) 3w 5k 3 00) Find the area. x 5 x 2 0) Find the area. x 3 x 4 w k 2 A 3w 2 sq units; P 8w units A 5k 5 sq units; P 0k 3 2k 2 units 02) Here are the shape and dimensions of the Millers family room. They will have wall-to-wall carpeting installed, and the carpet they have chosen costs \$2.50/ft 2. 3x 3 8 x2 sq units 5 4 x2 sq units 4x 4x a) Write an expression for the amount of carpet they will need. (Include the correct units.) 45 4 x2 ft 2 b) Write an expression for the cost of carpeting the family room. (Include the correct units.) x2 dollars 3 x 4 3 x 4 x x R) When can you distribute an exponent into parentheses? R2) When can you not distribute an exponent into parentheses? R3) When do you add the exponents? R4) When do you multiply the exponents? R5) Which exercises in this section do you find most challenging? Where could you go for help? 36 CHAPTER 5 Rules of Exponents

11 5.B Basic Rules of Exponents: Combining the Rules What is your objective for Section 5.B? Combine the Product Rule and Power Rules of Exponents How can you accomplish the objective? Be able to follow the order of operations correctly, and complete the given example on your own. Complete You Try. Read the explanations, follow the examples, take notes, and complete the You Try. Combine the Product Rule and Power Rules of Exponents When we combine the rules of exponents, we follow the order of operations. EXAMPLE In-Class Example Simplify. a) (4f) 2 (2f) 3 b) 3(4a 2 b 3 ) 2 c) (2y3 ) 2 (4z 0 ) 3 Answer: a) 28f 5 b) 48a 4 b 6 c) 9y 6 4z 30 Notice how the parentheses are used in these examples. Simplify. a) (2c) 3 (3c 8 ) 2 b) 2(5k 4 m 3 ) 3 c) (6t 5 ) 2 (2u 4 ) 3 Solution a) (2c) 3 (3c 8 ) 2 Because evaluating exponents comes before multiplying in the order of operations, evaluate the exponents first. (2c) 3 (3c 8 ) 2 (2 3 c 3 )(3 2 )(c 8 ) 2 Use the power rule. (8c 3 )(9c 6 ) Use the power rule and evaluate exponents. 72c 9 Product rule b) 2(5k 4 m 3 ) 3 Which operation should be performed first, multiplying 2 5 or simplifying (5k 4 m 3 ) 3? In the order of operations, we evaluate exponents before multiplying, so we will begin by simplifying (5k 4 m 3 ) 3. c) 2(5k 4 m 3 ) 3 2 (5) 3 (k 4 ) 3 (m 3 ) 3 Order of operations and power rule 2 25k 2 m 9 Power rule 250k 2 m 9 Multiply. (6t 5 ) 2 (2u 4 ) 3 What comes first in the order of operations, dividing or evaluating exponents? Evaluating exponents. (6t 5 ) 2 (2u 4 ) 36t0 3 8u 2 Power rule 36 9 t 0 u 8 2 9t0 2u 2 2 Divide out the common factor of 4. SECTION 5.B Basic Rules of Exponents: Combining the Rules 37

12 When simplifying the expression in Example c, (6t 5 ) 2 (2u 4 ) 3, it may be tempting to simplify before applying the product rule, like this: (6 3 t 5 ) 2 (2u 4 ) (3t 5 ) 2 0 9t 3 (u 4 3 Wrong! 2 ) u You can see, however, that because we did not follow the rules for the order of operations, we did not get the correct answer. [ YOU TRY ] Simplify. a) 4(2a 9 b 6 ) 4 b) (7x 0 y) 2 ( x 4 y 5 ) 4 0(m 2 n 3 ) 5 c) d) a 2 (5p 4 ) 2 6 w7 b (3w ) 3 ANSWERS TO [YOU TRY] EXERCISE ) a) 64a 36 b 24 b) 49x 36 y 22 c) 2m0 n 5 d) 3 5p 8 4 w47 5.B Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective : Combine the Product Rule and Power Rules of Exponents ) When evaluating expressions involving exponents, always keep in mind the order of. operations 2) The first step in evaluating (9 3) 2 is. subtraction Simplify. 3) (k 9 ) 2 (k 3 ) 2 k 24 4) (d 5 ) 3 (d 2 ) 4 d 23 5) (5z 4 ) 2 (2z 6 ) 3 200z 26 6) (3r) 2 (6r 8 ) 2 324r 8 7) 6ab( a 0 b 2 ) 3 6a 3 b 7 8) 5pq 4 ( p 4 q) 4 5p 7 q 8 9) (9 2) 2 2 0) (8 5) 3 27 ) ( 4t 6 u 2 ) 3 (u 4 ) 5 64t 8 u 26 2) ( m 2 ) 6 ( 2m 9 ) 4 6m 48 3) 8(6k 7 l 2 ) 2 288k 4 l 4 4) 5( 7c 4 d ) 2 245c 8 d 2 5) a 3 g 5b 3 a 6 b 2 3 4g 5 3 6) a 2 5 z5 b (0z) z7 7) a n2 b ( 4n 9 ) n22 8) a d8 b a d3 b 4d 38 9) h 4 (0h 3 ) 2 ( 3h 9 ) 2 900h 28 20) v 6 ( 2v 5 ) 5 ( v 4 ) 3 32v 43 2) 3w (7w 2 ) 2 ( w 6 ) 5 22) 5z 3 ( 4z) 2 (2z 3 ) 2 320z 47w 45 23) (2x3 ) x (0y 5 ) 24) ( 3a4 ) y 0 (6b) 2 3a 2 4b 2 (4d 9 ) 2 8 d 25) ( 2c 5 ) 26) ( 5m7 ) c 30 (5n 2 ) 2 5m n 24 27) 8(a4 b 7 ) 9 (6c) 2 29) r 4 (r 5 ) 7 2t(t 2 ) 2 3) a x3 yb a x6 y 4 b 2a 36 b 63 28) 9c 2 33) a c9 d 2 b a cd 6 b 34) 2 2 a3 2 m3 n 0 b 35) a 5x5 y 2 z 4 37) a 3t4 u 9 2v 7 b 4 39) a 2w5 4x 3 y 6b 2 (3x 5 ) 3 2(yz 2 ) 6 39 r 30) k 5 (k 2 ) 3 242t 5 7m 0 (2m 3 ) 2 5 9x 7y 6 z 2 k 28m 6 32) (6s 8 t 3 ) 2 a st4 b 3 x24 y 4 400s 8 t 4 0 c29 d m6 n x 5 y 6 b 36) a 7a4 2 b z 2 8c b 6 8t 6 u ) a2pr8 28 6v q b 9w 0 40) c 5 2 x 6 y 2 a0b3 5a b 49a 8 b 2 64c 2 32p 5 r 40 q 55 4b 6 c 0 9a 2 38 CHAPTER 5 Rules of Exponents

13 Determine whether the equation is true or false. 4) (2k 2 2k 2 ) (2k 2 ) 2 false 42) (4c 3 2c 2 ) 2c false For Exercises 43 and 44, answer always, sometimes, or never. 43) If a and b are any integer values except zero, then the exponential expression b 2 ( a) 4 will always, sometimes, or never give a positive result. never 44) If a and b are any integer values except zero, then the exponential expression a a 3 b b will always, sometimes, or never give a positive result. sometimes 45) The length of a side of a square is 5l 2 units. a) Write an expression for its perimeter. 20l 2 units b) Write an expression for its area. 25l 4 sq units 46) The width of a rectangle is 2w units, and the length of the rectangle is 7w units. a) Write an expression for its area. 4w 2 sq units b) Write an expression for its perimeter. 8w units 47) The length of a rectangle is x units, and the width of the rectangle is 3 8 x units. 3 a) Write an expression for its area. 8 x2 sq units b) Write an expression for its perimeter. 4 x units 48) The width of a rectangle is 4y 3 units, and the length of the rectangle is 3 2 y3 units. a) Write an expression for its perimeter. 2y 3 units b) Write an expression for its area. 26y 6 sq units R) How are the parentheses being used in these problems? R2) Which exercises do you need to come back to and try again? How will this help you prepare for an exam? 5.2A Integer Exponents: Real-Number Bases What are your objectives for Section 5.2A? Use 0 as an Exponent 2 Use Negative Integers as Exponents How can you accomplish each objective? Understand the definition of zero as an exponent and write it in your notes. Complete the given example on your own. Complete You Try. Understand the definition of negative exponent and write it in your notes. Complete the given example on your own. Complete You Try 2. Read the explanations, follow the examples, take notes, and complete the You Trys. SECTION 5.2A Integer Exponents: Real-Number Bases 39

14 So far, we have defined an exponential expression such as 2 3. The exponent of 3 indicates that (3 factors of 2) so that Is it possible to have an exponent of zero or a negative exponent? If so, what do they mean? Use 0 as an Exponent Definition Zero as an Exponent: If a 0, then a 0. How can this be possible? Let s evaluate Using the product rule, we get: But we know that Therefore, if , then 2 0. This is one way to understand that a 0. EXAMPLE In-Class Example Evaluate. a) 3 0 b) 6 0 c) ( 4) 0 d) 2 0 (4) Answer: a) b) c) d) 4 Evaluate each expression. a) 5 0 b) 8 0 c) ( 7) 0 d) 3(2 0 ) Solution a) 5 0 b) c) ( 7) 0 d) 3(2 0 ) 3() 3 [ YOU TRY ] Evaluate. a) 9 0 b) 2 0 c) ( 5) 0 d) 3 0 ( 2) 2 Use Negative Integers as Exponents So far, we have worked with exponents that are zero or positive. What does a negative exponent mean? Let s use the product rule to find : ( 3) 2 0 Remember that a number multiplied by its reciprocal is, and here we have that a quantity, 2 3, times another quantity, 2 3, is. Therefore, 2 3 and 2 3 are reciprocals! This leads to the definition of a negative exponent. Did you notice that the signs of the bases do NOT change? Definition Negative Exponent: If n is any integer and a and b are not equal to zero, then a n a n a b n a n and aa b b a b n. a b Therefore, to rewrite an expression of the form a n with a positive exponent, take the reciprocal of the base and make the exponent positive. 320 CHAPTER 5 Rules of Exponents

15 EXAMPLE 2 In-Class Example 2 Evaluate. a) 3 3 b) a b c) a 2 3 b d) ( 2) 2 Answer: a) c) 9 d) 4 27 b) Evaluate each expression. a) 2 3 b) a b Solution c) a 5 b 3 a) 2 3 : The reciprocal of 2 is 2, so 2 3 a 2 b 3 d) ( 7) b) a 3 2 b 4 : The reciprocal of 3 2 is 2 3, so a3 2 b 4 a 2 3 b Notice that a negative exponent does not make the answer negative! Before working out these examples, try to determine by inspection whether the answer is going to be positive or negative. c) a 5 b 3 : The reciprocal of 5 is 5, so a 5 b d) ( 7) 2 : The reciprocal of 7 is 7, so ( 7) 2 a 2 7 b a 2 7 b ( ) 2 a 2 7 b [ YOU TRY 2 ] Evaluate. a) (0) 2 b) a 4 b 2 c) a 2 3 b 3 d) 5 3 ANSWERS TO [YOU TRY] EXERCISES ) a) b) c) d) 2 2) a) 00 b) 6 c) 27 8 d) A Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Mixed Exercises: Objectives and 2 ) True or False: Raising a positive base to a negative exponent will give a negative result. (Example: 2 4 ) false 2) True or False: 8 0. true 3) True or False: The reciprocal of 4 is 4. true 4) True or False: false ) (5) 0 ( 5) 0 2 2) a b a b 3) ) ) ) a 2 8 b 4) ) 2 2 8) ) a 0 b Evaluate. 5) 2 0 6) ( 4) 0 2) a 2 b ) a 4 b 2 6 7) 5 0 8) 0 9) ) ( 9) 0 23) a 4 3 b ) a 2 5 b SECTION 5.2A Integer Exponents: Real-Number Bases 32

16 25) a 9 7 b 2 27) a 4 b 3 29) a 3 8 b ) a 0 3 b ) a 2 b ) a 5 2 b ) ) ) 5 34) ) ) ) ) ) ( 7) ) R) When evaluating an expression with negative exponents, when do you get a negative answer? R2) Explain the importance of fully understanding and using the order of operations in this chapter. 5.2B Integer Exponents: Variable Bases What are your objectives for Section 5.2B? Use 0 as an Exponent 2 Rewrite an Exponential Expression with Positive Exponents How can you accomplish each objective? Use the same definition of zero as an exponent used in the previous section. Complete the given example on your own. Complete You Try. Follow the explanation to understand how to rewrite an expression with positive exponents. Learn and apply the definition of a m bn n b a m. Complete the given examples on your own. Complete You Trys 2 and 3. Read the explanations, follow the examples, take notes, and complete the You Trys. Use 0 as an Exponent We can apply 0 as an exponent to bases containing variables. EXAMPLE In-Class Example Evaluate. Assume that the variable does not equal zero. a) k 0 b) ( m) 0 c) 3q 0 Answer: a) b) c) 3 Evaluate each expression. Assume that the variable does not equal zero. a) t 0 b) ( k) 0 c) (p) 0 Solution a) t 0 b) ( k) 0 c) (p) 0 (p) CHAPTER 5 Rules of Exponents

17 [ YOU TRY ] Evaluate. Assume that the variable does not equal zero. a) p 0 b) (0x) 0 c) (7s) 0 2 Rewrite an Exponential Expression with Positive Exponents Next, let s apply the definition of a negative exponent to bases containing variables. As in Example, we will assume that the variable does not equal zero since having zero in the denominator of a fraction will make the fraction undefined. Recall that 2 4 a 4 2 b. That is, to rewrite the expression with a positive 6 exponent, we take the reciprocal of the base. What is the reciprocal of x? The reciprocal is x. EXAMPLE 2 In-Class Example 2 Rewrite the expression with positive exponents. Assume that the variable does not equal zero. a) y 4 b) a 3 4 d b c) 7f 7 Answer: a) c) 7 7 f 4 y b) d 4 8 Before working out these examples, be sure you correctly identify the base. Rewrite the expression with positive exponents. Assume that the variable does not equal zero. a) x 6 b) a 2 6 n b c) 3a 2 Solution a) x 6 a x b 6 c) 3a 2 3 a a b 2 6 x 6 x 6 3 a 2 3 a 2 6 b) a2 n b a n 6 2 b n6 2 6 n6 64 The reciprocal of 2 n is n 2. Remember, the base is a, not 3a, since there are no parentheses. Therefore, the exponent of 2 applies only to a. [ YOU TRY 2 ] Rewrite the expression with positive exponents. Assume that the variable does not equal zero. a) m 4 b) a 7 z b c) 2y 3 How could we rewrite x 2 with only positive exponents? One way would be to apply 2 y x 2 2 the power rule for exponents: y ax 2 y b a y 2 x b y2 x 2 SECTION 5.2B Integer Exponents: Variable Bases 323

18 Notice that to rewrite the original expression with only positive exponents, the terms with the negative exponents switch their positions in the fraction. We can generalize this way: Definition If m and n are any integers and a and b are real numbers not equal to zero, then a m bn n b a m EXAMPLE 3 In-Class Example 3 Rewrite the expression with positive exponents. Assume that the variables do not equal zero. a) f 9s 0 b) c) c 2 d g t Answer: a) g5 f c) c 2 7 d b) 9 s 0 4 t If the exponent is already positive, do not change the position of the expression in the fraction. [ YOU TRY 3 ] Rewrite the expression with positive exponents. Assume that the variables do not equal zero. c 8 a) 3 d b) 5p 6 q c) 2xy t 2 u d) e) aab 2 3z 4c b Solution a) b) c 8 3 c 8 3 d d 5p 6 q 7 5 p 6 q 7 c) t 2 u t 2 u t 2 u t 2 u d) 2xy 3 3z 2 2xz2 3y 3 e) a ab 4c b 3 a 4c ab b 3 43 c 3 a 3 b 3 To make the exponents positive, switch the positions of the terms in the fraction. Since the exponent on q is positive, we do not change its position in the expression. Move t 2 u to the denominator to write with positive exponents. To make the exponents positive, switch the positions of the factors with negative exponents in the fraction. To make the exponent positive, use the reciprocal of the base. Power rule 64c3 a 3 b 3 Simplify. Rewrite the expression with positive exponents. Assume that the variables do not equal zero. a) n 6 y b) z 9 2 3k c) 4 8x 5 y d) 4 2 8d e) 6m 2 a3x2 n y b ANSWERS TO [YOU TRY] EXERCISES ) a) b) c) 2) a) m b) 4 z7 c) 2 2 y 3) a) 3 y n 6 b) c) 8y 2 x d) 4n 5 3m 2 d e) y 4 9x 4 k 4 3z CHAPTER 5 Rules of Exponents

19 5.2B Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective : Use 0 as an Exponent ) Identify the base in each expression. a) w 0 w b) 3n 5 n c) (2p) 3 2p d) 4c 0 c 2) True or False: (6 4) 0 false Evaluate. Assume that the variables do not equal zero. 3) r 0 4) (5m) 0 5) 2k 0 2 6) z 0 7) x 0 (2x) 0 2 8) a b a b Objective 2: Rewrite an Exponential Expression with Positive Exponents Rewrite each expression with only positive exponents. Assume that the variables do not equal zero. 9) d 3 3 0) y 7 d y 7 ) p 2) a 5 p a 5 3) a 0 3 b h 2 k 4) b 3 a 0 k h 2 5) y 8 x 5 7) t 5 8u 3 5 x y 8 t 5 u 3 8 v 2 6) w 7 7 w v 2 8) 9x 4 y 5 9 x 4 y 5 9) 5m 6 n 2 5m 6 20) n 2 9 a 4 b 3 b 3 9a 4 2) 2 t u 5 2t u 5 22) 23) 8a6 b 5c 0 d 25) 8a 6 c 0 5bd 24) 2z 4 x 7 y 6 2x7 y 6 z 4 26) 27) a a 6 b 2 29) a 2n q b 5 36 a 2 q 5 32n 5 7r 2t 9 u 2 7k 8 h 5 20m 7 n 2 9 7rt 2u 2 0 7h 5 m 7 n 2 20k 8 a 2 b 2 c a2 b 2 c 28) a3 y b 4 30) a w 5v b 3 y v 3 w 3 3) a 2b cd b 2 c 2 2 d 44b ) a2tu v b 33) 9k 2 9 k 2 34) 3g ) 3t ) 8h 4 8 t 3 h 4 37) m 9 38) d 5 m 9 5 d 39) a z b 0 4) a j b 43) 5 a n b 2 45) c a d b 3 j z 0 5n 2 cd 3 40) a k b 6 42) a c b 7 44) 7 a t b 8 g 5 46) x 2 a y b 2 k 6 c 7 v 6 64t 6 u 6 7t 8 x 2 y 2 For Exercises 47 50, answer always, sometimes, or never. 47) If a is any integer value except zero, then the exponential expression a 2 will always, sometimes, or never give a negative result. always 48) If b is any integer value except zero, then the exponential expression ( b) 3 will always, sometimes, or never give a positive result. sometimes 49) If a and b are any integer values except zero, then the exponential expression a 0 b 0 will always, sometimes, or never equal zero. always 50) If a and b are any integer values except zero, then the exponential expression (a 0 b 0 ) 2 will always, sometimes, or never equal zero. never Determine whether the equation is true or false. 5) 2 2t 5 t 5 true 52) 6x5 7 6x x 3 false 53) (p 3q ) 2 a q 3p b 2 54) (h 2 4k 2 ) 2 a 2h k b 4 false true SECTION 5.2B Integer Exponents: Variable Bases 325

20 R) If a variable is raised to the power zero, the answer will always be what number? Assume the variable does not equal zero. R2) Why is it useful to write expressions without negative exponents? R3) Were there any problems you were unable to do? If so, write them down or circle them and ask your instructor for help. 5.3 The Quotient Rule What is your objective for Section 5.3? Use the Quotient Rule for Exponents How can you accomplish the objective? Learn the property for the Quotient Rule for Exponents and write an example in your notes. Complete the given examples on your own. Complete You Trys and 2. Read the explanations, follow the examples, take notes, and complete the You Trys. Use the Quotient Rule for Exponents In this section, we will discuss how to simplify the quotient of two exponential expressions with the same base. Let s begin by simplifying One way to simplify this expression is to write the numerator and denominator without exponents: 8 6 Therefore, Divide out common factors. Do you notice a relationship between the exponents in the original expression and the exponent we get when we simplify? That s right. We subtracted the exponents CHAPTER 5 Rules of Exponents

21 Property Quotient Rule for Exponents If m and n are any integers and a 0, then a m a n am n To divide expressions with the same base, keep the base the same and subtract the denominator s exponent from the numerator s exponent. EXAMPLE In-Class Example Simplify. Assume that the variables do not equal zero. a) 74 n2 b) 2 7 n c) d) x3 43 e) 5 x 4 6 Answer: a) 49 b) n 8 c) 6 d) 2 x e) 64 Be careful when you subtract exponents, especially when you are working with negative numbers! Simplify. Assume that the variables do not equal zero. 2 9 a) 2 b) t 0 3 t c) d) n 5 2 n e) Solution a) b) c) d) Since the bases are the same, subtract the exponents. t 0 t 4 t0 4 t 6 Because the bases are the same, subtract the exponents ( 2) Since the bases are the same, subtract the exponents Be careful when subtracting the negative exponent! n 5 n 7 n5 7 n 2 Same base; subtract the exponents. a 2 n b Write with a positive exponent. 2 n [ YOU TRY ] e) Simplify. Assume that the variables do not equal zero. 5 7 a) 5 b) c 4 4 c c) k 2 k d) Because the bases are not the same, we cannot apply the quotient rule. Evaluate the numerator and denominator separately. We can apply the quotient rule to expressions containing more than one variable. Here are more examples: EXAMPLE 2 In-Class Example 2 Simplify. Assume that the variables do not equal zero. a) f 4 g 6p 6 q 4 b) 3 fg 6p 2 q 3 c) 30a2 b 6 48a 4 b 4 Simplify. Assume that the variables do not equal zero. x 8 y 7 a) x 3 y b) 2a 5 b 0 4 8a 3 b 2 Solution x 8 y 7 a) x 3 y 4 x8 3 y 7 4 x 5 y 3 Subtract the exponents. SECTION 5.3 The Quotient Rule 327

22 Answer: a) f 3 g 8 5 c) 8a 2 b 0 b) 8q 3p 4 b) 2a 5 b 0 8a 3 b 2 We will reduce 2 in addition to applying the quotient rule a 5 b 0 3 8a 3 b 2 2 a 5 ( 3) b 0 2 Subtract the exponents a 5 3 b a 2 b 8 3b8 2a 2 [ YOU TRY 2 ] Simplify. Assume that the variables do not equal zero. a) r 4 s 0 rs 3 b) 30m 6 n 8 42m 4 n 3 ANSWERS TO [YOU TRY] EXERCISES ) a) 25 b) c 5 c) k d) 8 6 2) a) r 3 s 7 b) 5m2 7n Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective : Use the Quotient Rule for Exponents State what is wrong with the following steps and then simplify correctly. ) a5 a 3 a3 5 a 2 You must subtract the a 2 denominator s exponent from the numerator s exponent; a 2. 2) a4 2 b You must have the same base in order to use the quotient rule;. Determine whether the equation is true or false. 3) false 4) true 5) t 7 t 5 t 2 false 6) m 0 m 3 m 7 false 7) r 6 r 2 r 4 true 8) k 5 k 0 k 5 false Simplify using the quotient rule. Assume that the variables do not equal zero. 9) d 0 5 d d 5 0) z z 7 z4 ) m9 m 5 m4 3) 8t5 t 8 8t7 2) a6 a a 5 4) 4k4 k 2 4k2 5) ) ) ) ) ) ) ) ) 0d 4 d 2 0d 2 24) 3x6 x 2 3x4 25) 20c 30c c5 26) 35t7 56t t5 27) y3 y 8 y 5 29) x 3 x 6 x 9 3) t 6 t 3 t 3 33) a a 9 a 0 35) t4 t 37) 5w2 5 w 0 w 8 t 3 36) m4 28) m 0 m 6 u 20 30) u 9 u y8 32) y 5 y 7 m 9 34) m 3 m 6 c 7 c8 c 7p3 38) p 2 7 p CHAPTER 5 Rules of Exponents

23 39) 6k k 4 6 2h3 2 40) 3 k h 7 h 4 4) a4 b 9 ab 2 a3 b 7 42) p5 q 7 43) 0k 2 6 l 5k 5 l 2 3 2k 8 3l p 2 q 3 p3 q 4 44) 28tu 2 4t 5 u 9 7 2u 45) 300x7 y 3 30x 2 y 8 0 x 5 y 5 46) 63a 3 b 2 7a 7 b 8 9 a 0 b 6 t 4 47) 6v w 54v 2 w 5 6 w 9v 3 48) 3a 2 b 8a 0 b 6 2 a 6b 7 49) 3c5 2 d 3 3 8cd 8 c4 d 50) 9x 5 y 2 9 4x 2 y 6 4x 3 y 4 5) 53) (x y)9 (x y) 2 (x y)7 52) (c d ) 5 (c d ) (c d )6 54) (a b)9 (a b)5 4 (a b) (a 2b) 3 a 2b 4 (a 2b) R) How could you change a positive exponent to a negative exponent? R2) When do you add exponents? R3) When do you subtract exponents? R4) When do you multiply exponents? R5) In what other courses have you seen exponents used? Putting It All Together What is your objective? Combine the Rules of Exponents How can you accomplish the objective? Understand all of the Rules of Exponents and write the summary in your notes. Complete the given example on your own. Complete You Try. Read the explanations, follow the examples, take notes, and complete the You Trys. Summarize all of the rules of exponents in your notes. Combine the Rules of Exponents Let s see how we can combine the rules of exponents to simplify expressions. Putting It All Together 329

24 EXAMPLE In-Class Example Simplify using the rules of exponents. Assume that all variables represent nonzero real numbers. a) (4h 3 ) 2 (2h) 3 b) z9 z 6 c) a 2p 7 q 3 28p 8 q 4b 2 z 4 Answer: a) 28 h b) 3 z c) 6p30 9q 4 Remember to use the order of operations when simplifying expressions like these. Simplify using the rules of exponents. Assume that all variables represent nonzero real numbers. a) (2t 6 ) 3 (3t 2 ) 2 w 3 w 4 b) c) a 2a 2 b 9 3 w 6 30ab b 2 Solution a) (2t 6 ) 3 (3t 2 ) 2 We must follow the order of operations. Therefore, evaluate the exponents first. b) (2t 6 ) 3 (3t 2 ) t ( 6)(3) 3 2 t (2)(2) Apply the power rule. 8t 8 9t 4 Simplify. 72t 8 4 Multiply 8 9 and add the exponents. 72t 4 72 t 4 Write the answer using a positive exponent. w 3 w 4 w 6 Let s begin by simplifying the numerator: w 3 w 4 w 3 4 w 6 w 6 Add the exponents in the numerator. w w 6 Now, we can apply the quotient rule: w 6 w 5 Subtract the exponents. w 5 Write the answer using a positive exponent. c) a 2a 2 b 9 30ab 2 b 3 Eliminate the negative exponent outside the parentheses by taking the reciprocal of the base. Notice that we have not eliminated the negatives on the exponents inside the parentheses. a 2a 2 b ab b a 30ab a 2 b 9b Next, we could apply the exponent of 3 to the quantity inside the parentheses. However, we could also simplify the expression inside the parentheses before cubing it. a 30ab 2 3 2a 2 b 9b a a 22 b 2 9 b a a3 b b Simplify 30 and subtract the exponents a9 b 33 Apply the power rule. 25a9 8b 33 Write the answer using positive exponents. 330 CHAPTER 5 Rules of Exponents

25 [ YOU TRY ] Simplify using the rules of exponents. a) a m2 n 3 m 4 n b 4 b) ( p 5 ) 4 (6p 7 ) 2 c) a 9x4 y 5 54x 3 y b 2 If variables appear in the exponents, the same rules apply. ANSWERS TO [YOU TRY] EXERCISES ) a) m 32 n 8 b) 36 p 6 c) 36y 2 x 2 Putting It All Together Exercises *Additional answers can be found in the Answers to Exercises appendix. Objective : Combine the Rules of Exponents Use the rules of exponents to evaluate. ) a 2 3 b ) (2 2 ) ) ) ( 5)6 ( 5) 2 ( 5) ) a 0 3 b ) a 3 7 b 2 7) (9 6) 2 9 8) (3 8) ) ) ) a b a b 5) ) ) ) a 2 8 b ) Simplify. Assume that all variables represent nonzero real numbers. The final answer should not contain negative exponents. 7) 0( 3g 4 ) 3 270g 2 8) 7(2d 3 ) 3 56d 9 9) 33s s 2 33 s 2) a 2xy4 3x 9 y 2b 4 c 7 20) c 2 c x40 y 24 22) a a6 b 5 3 0a 3b a 9 b Do the exercises, and check your work. 23) a 9m8 2 n b n ) 3 8m 6 a3s 6 r b 2 25) ( b 5 ) 3 b 5 26) (h ) 8 h 88 r 8 s ) ( 3m 5 n 2 ) 3 27m 5 n 6 28) (3a 6 b) 2 69a 2 b 2 29) a 9 4 z5 ba 8 3 z 2 b 6z 3 30) (5w 3 )a 3 5 w6 b 9w 9 3) a s7 t 3 b 6 t 8 s 42 33) ( ab 3 c 5 ) 2 a a4 bc b 3 35) a 48u 7 v 2 36u 3 v 5b 3 37) a 3t4 u t 2 u 4 b 3 m 3 32) n 4 m 3 n 4 a 4 b 3 c 7 34) (4v3 ) 2 4 (6v 8 ) 2 9v 0 27u 30 64v 2 2 xy5 36) a 9x 2 y b 27t 6 u 5 38) a k7 m 7 2k m 6b 2 8 x 6 y 8 39) (h 3 ) 6 40) ( d 4 ) 5 h 8 20 d 4) a h 4 2 b h ) 3f 2 3 f 2 43) 7c 4 ( 2c 2 ) 3 56c 0 44) 5p 3 (4p 6 ) 2 80p 5 45) (2a 7 ) (6a) ) a 5 (9r2 s 2 ) 9r 2 s 2 47) a 9 20 r4 b(4r 3 )a 2 33 r9 b 48) a f 8 f 3 6 f 2 f b 9 49) (a2 b 5 c) 3 a 2 b 9 (a 4 b 3 c) 2 c 6 55 r0 50) (x y 7 z 4 ) 3 x (x 4 yz 5 ) 3 44 k6 m 2 f 36 9 y 24 z 3 Putting It All Together 33

27 Read the explanations, follow the examples, take notes, and complete the You Trys. The distance from Earth to the sun is approximately 50,000,000 km. A single rhinovirus (cause of the common cold) measures mm across. Performing operations on very large or very small numbers like these can be difficult. This is why scientists and economists, for example, often work with such numbers in a shorthand form called scientific notation. Writing numbers in scientific notation together with applying rules of exponents can simplify calculations with very large and very small numbers. Multiply a Number by a Power of 0 Before discussing scientific notation further, we need to understand some principles behind the notation. Let s look at multiplying numbers by positive powers of 0. EXAMPLE In-Class Example Multiply. a) b) Answer: a) 95 b) 327. Multiply. a) b) Solution a) b) Notice that when we multiply each of these numbers by a positive power of 0, the result is larger than the original number. In fact, the exponent determines how many places to the right the decimal point is moved. [ YOU TRY ] place to the right 3 places to the right Multiply by moving the decimal point the appropriate number of places. a) b) What happens to a number when we multiply by a negative power of 0? EXAMPLE 2 In-Class Example 2 Multiply. a) b) Answer: a) b) Multiply. a) b) Solution a) b) , , When we multiply each of these numbers by a negative power of 0, the result is smaller than the original number. The exponent determines how many places to the left the decimal point is moved: places to the left 4 places to the left SECTION 5.4 Scientific Notation 333

28 [ YOU TRY 2 ] Multiply. a) b) It is important to understand the previous concepts to understand how to use scientific notation. 2 Understand Scientific Notation Definition A number is in scientific notation if it is written in the form a 0 n, where a 0 and n is an integer. Note Multiplying a by a positive power of 0 will result in a number that is larger than a. Multiplying a by a negative power of 0 will result in a number that is smaller than a. The double inequality a 0 means that a is a number that has one nonzero digit to the left of the decimal point. In other words, a number in scientific notation has one digit to the left of the decimal point and the number is multiplied by a power of 0. Here are some examples of numbers written in scientific notation: ,.2 0 3, and The following numbers are not in scientific notation: c c c 2 digits to left Zero is to left 3 digits to left of decimal point of decimal point of decimal point Now let s convert a number written in scientific notation to a number without exponents. EXAMPLE 3 In-Class Example 3 Rewrite without exponents. a) b) c) Answer: a) 745 b) c) 39,040 How does this process compare to Examples and 2? Rewrite without exponents. a) b) c) Solution a) S ,230 Remember, multiplying by a positive power of 4 places to the right 0 will make the result larger than b) S Multiplying by a negative power of 0 will 3 places to the left make the result smaller than 7.4. c) S places to the right 334 CHAPTER 5 Rules of Exponents

29 [ YOU TRY 3 ] Rewrite without exponents. a) b) c) Write a Number in Scientific Notation Let s write the number 48,000 in scientific notation. First locate its decimal point: 48,000. Next, determine where the decimal point will be when the number is in scientific notation: 48,000. Decimal point will be here. Therefore, 48, n, where n is an integer. Will n be positive or negative? We can see that 4.8 must be multiplied by a positive power of 0 to make it larger, 48, a Decimal point Decimal point will be here. starts here. Now we count four places between the original and the final decimal place locations We use the number of spaces, 4, as the exponent of 0. 48, EXAMPLE 4 In-Class Example 4 Write each number in scientific notation. a) In 200, New York City s population was approximately 8,200,000 people. b) The speed of a garden snail is 0.03 mph. Answer: a) people b) mph Write each number in scientific notation. Solution a) The distance from Earth to the sun is approximately 50,000,000 km. 50,000, ,000,000. Move decimal point 8 places. a Decimal point Decimal point will be here. is here. 50,000,000 km km b) A single rhinovirus measures mm across mm mm mm Decimal point will be here. SECTION 5.4 Scientific Notation 335

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