You evaluated powers. You will simplify expressions involving powers. Consider what happens when you multiply two powers that have the same base:

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1 TEKS.1 a.1, 2A.2.A Before Now Use Properties of Eponents You evaluated powers. You will simplify epressions involving powers. Why? So you can compare the volumes of two stars, as in Eample. Key Vocabulary scientific notation Consider what happens when you multiply two powers that have the same base: 2 p 2 (2 p 2 p 2) p (2 p 2 p 2 p 2 p 2) 2 8 Note that the eponent 8 in the product is the sum of the eponents and in the factors. This property is one of several properties of eponents shown below. KEY CONCEPT For Your Notebook Properties of Eponents Let a and b be real numbers and let m and n be integers. AVOID ERRORS When you multiply powers, do not multiply the bases. For eample, 2 p Þ 9. Property Name Definition Eample Product of Powers a m p a n a m 1 n p 21 1 (21) 2 2 Power of a Power (a m ) n a mn ( ) 2 p Power of a Product (ab) m a m b m (2 p ) p Negative Eponent a 2m 1 a m,aþ Zero Eponent a 0 1, a Þ 0 (289) 0 1 Quotient of Powers a m a n am 2 n, a Þ (26) Power of a Quotient 1 a b 2 m am b m, b Þ E XAMPLE 1 Evaluate numerical epressions a. (24 p 2 ) 2 (24) 2 p (2 ) 2 Power of a product property 16 p 2 p 2 Power of a power property 16 p ,84 Simplify and evaluate power. b Negative eponent property Quotient of powers property Simplify and evaluate power. 0 Chapter Polynomials and Polynomial Functions

2 SCIENTIFIC NOTATION A number is epressed in scientific notation if it is in the form c 10 n where 1 c < 10 and n is an integer. When you work with numbers in scientific notation, the properties of eponents can make calculations easier. E XAMPLE 2 Use scientific notation in real life LOCUSTS A swarm of locusts may contain as many as 8 million locusts per square kilometer and cover an area of 1200 square kilometers. About how many locusts are in such a swarm? Solution Number of locusts Locusts per square kilometer Number of square kilometers REVIEW SCIENTIFIC NOTATION For help with scientific notation, see p ,000, Substitute values. (8. 10 )( ) Write in scientific notation. (8. 1.2)(10 10 ) Use multiplication properties Product of powers property Write 10.2 in scientific notation Product of powers property c The number of locusts is about , or about 102,000,000,000. GUIDED PRACTICE for Eamples 1 and 2 Evaluate the epression. Tell which properties of eponents you used. 1. (4 2 ) 2. (28)(28) p p 10 SIMPLIFYING EXPRESSIONS You can use the properties of eponents to simplify algebraic epressions. A simplified epression contains only positive eponents. E XAMPLE Simplify epressions a. b 24 b 6 b b b 9 Product of powers property INTERPRET BASES In this book, it is assumed that any base with a zero or negative eponent is nonzero. b. 1 r22 s 2 2 (r22 ) 2 (s ) 2 Power of a quotient property r6 s 29 Power of a power property r 6 s 9 Negative eponent property c. 16m 4 n 2 2n 2 8m 4 n 2 2 (2) Quotient of powers property 8m 4 n 0 8m 4 Zero eponent property at classzone.com.1 Use Properties of Eponents 1

3 E XAMPLE 4 TAKS PRACTICE: Multiple Choice What is the simplified form of (22 y ) 4 y 9? A 2 y B 1 10 C 1 2 y D 1 10 y Solution ( 22 y ) (22 ) (y ) 4 y 9 Power of a product property 4 y 9 26 y 9 4 y 9 Power of a power property y Quotient of powers property 210 y 0 Simplify eponents. 210 p 1 Zero eponent property 1 10 Negative eponent property c The correct answer is B. A B C D E XAMPLE Compare real-life volumes ASTRONOMY Betelgeuse is one of the stars found in the constellation Orion. Its radius is about 100 times the radius of the sun. How many times as great as the sun s volume is Betelgeuse s volume? Solution Let r represent the sun s radius. Then 100r represents Betelgeuse s radius. 4 Betelgeuse s volume π(100r) The volume of a sphere is 4 pr. Sun s volume 4 πr 4 π100 r Power of a product property 4 πr 100 r 0 Quotient of powers property 100 p 1 Zero eponent property,,000,000 Evaluate power. c Betelgeuse s volume is about.4 billion times as great as the sun s volume. 2 Chapter Polynomials and Polynomial Functions

4 GUIDED PRACTICE for Eamples, 4, and Simplify the epression. Tell which properties of eponents you used (y 2 z )(y 24 z 21 ). 1 s t y 22 y EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS on p. WS1 for Es. 1, 1, and 1 TAKS PRACTICE AND REASONING Es. 6, 46, 1,,, and 6 1. VOCABULARY State the name of the property illustrated. a. a m p a n a m 1 n b. a 2m 1 a m, a? 0 c. (ab)m a m b m 2. WRITING Is the number in scientific notation? Eplain. EXAMPLE 1 on p. 0 for Es. 14 EVALUATING NUMERICAL EXPRESSIONS Evaluate the epression. Tell which properties of eponents you used.. p 2 4. (4 22 ). (2)(2) 4 6. (2 4 ) p p 6 0 p EXAMPLE 2 on p. 1 for Es. 1 2 SCIENTIFIC NOTATION Write the answer in scientific notation. 1. ( )( ) 16. ( )( ) 1. (6. 10 )( ) 18. ( )( ) 19. ( ) 20. ( ) ( )( ) EXAMPLES and 4 on pp. 1 2 for Es SIMPLIFYING ALGEBRAIC EXPRESSIONS Simplify the epression. Tell which properties of eponents you used. 24. w 22 w 6 2. (2 2 y ) 26. (p q 2 ) (w 22 )(w 6 21 ) 28. (s 22 t 4 ) (a b ) y 2 2 y c d 9cd r 4 s 24r 4 s 2. 2a b 24 a b y 11 4z p 8z y. 2 y 2 y 2 p y TAKS RL EASONING 2 2 y What is the simplified form of? 6y 21 A y 2 B y 2 C D 1.1 Use Properties of Eponents

5 ERROR ANALYSIS Describe and correct the error in simplifying the epression p 1 9. (2) 2 (2) GEOMETRY Write an epression for the figure s area or volume in terms of. 40. A Ï 4 s V πr 2 h 42. V lwh 2 2 REASONING Write an epression that makes the statement true y 12 z 8 4 y z 11 p? 44. y y? 4. (a b 4 ) 2 a 14 b 21 p? 46. TAKS REASONING Find three different ways to complete the following statement so that it is true: 12 y 16 (? y? )(? y? ). CHALLENGE Refer to the properties of eponents on page Show how the negative eponent property can be derived from the quotient of powers property and the zero eponent property. 48. Show how the quotient of powers property can be derived from the product of powers property and the negative eponent property. PROBLEM SOLVING EXAMPLE 2 on p. 1 for Es OCEAN VOLUME The table shows the surface areas and average depths of four oceans. Calculate the volume of each ocean by multiplying the surface area of each ocean by its average depth. Write your answers in scientific notation. Ocean Surface area (square meters) Average depth (meters) Pacific Atlantic Indian Arctic EARTH SCIENCE The continents of Earth move at a very slow rate. The South American continent has been moving about mile per year for the past 12,000,000 years. How far has the continent moved in that time? Write your answer in scientific notation. 4 WORKED-OUT SOLUTIONS on p. WS1 TAKS PRACTICE AND REASONING

6 EXAMPLE on p. 2 for Es TAKS REASONING A typical cultured black pearl is made by placing a bead with a diameter of 6 millimeters inside an oyster. The resulting pearl has a diameter of about 9 millimeters. Compare the volume of the resulting pearl with the volume of the bead. 2. MULTI-STEP PROBLEM A can of tennis balls consists of three spheres of radius r stacked vertically inside a cylinder of radius r and height h. a. Write an epression for the total volume of the three tennis balls in terms of r. b. Write an epression for the volume of the cylinder in terms of r and h. c. Write an epression for h in terms of r using the fact that the height of the cylinder is the sum of the diameters of the three tennis balls. d. What fraction of the can s volume is taken up by the tennis balls?. TAKS REASONING You can think of a penny as a cylinder with a radius of about 9. millimeters and a height of about 1. millimeters. a. Calculate Approimate the volume of a penny. Give your answer in cubic meters. b. Estimate Approimate the volume of your classroom in cubic meters. Eplain how you obtained your answer. c. Interpret Use your results from parts (a) and (b) to estimate how many pennies it would take to fill your classroom. Do you think your answer is an overestimate or an underestimate? Eplain. 4. CHALLENGE Earth s core is approimately spherical in shape and is divided into a solid inner core (the yellow region in the diagram shown) and a liquid outer core (the dark orange region in the diagram). a. Earth s radius is about times as great as the radius of Earth s inner core. Find the ratio of Earth s total volume to the volume of Earth s inner core. b. Find the ratio of the volume of Earth s outer core to the volume of Earth s inner core. MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 4.4; TAKS Workbook. TAKS PRACTICE What are the zeros of the function y ? TAKS Obj. A 2 2, 24 B 2 2, 4 C 2, 24 D 2, 4 REVIEW Skills Review Handbook p. 994; TAKS Workbook 6. TAKS PRACTICE In the diagram, ] NP bisects MNQ and m MNP is 8. Which equation can be used to find y, which represents m MNQ? TAKS Obj. 6 F y 2 G y M N 8 P P H y 2 J y EXTRA PRACTICE for Lesson.1, p ONLINE QUIZ at classzone.com

7 Investigating g Algebra ACTIVITY Use before Lesson.2.2 End Behavior of Polynomial Functions MATERIALS graphing calculator TEKS a., a.6, 2A.4.A; P.1.E TEXAS classzone.com Keystrokes QUESTION How is the end behavior of a polynomial function related to the function s equation? Functions of the form f() 6 n, where n is a positive integer, are eamples of polynomial functions. The end behavior of a polynomial function s graph is its behavior as approaches positive infinity (1`) or as approaches negative infinity (2`). E XPLORE Investigate the end behavior of f() 6 n where n is even Graph the function. Describe the end behavior of the graph. a. f() 4 b. f() 2 4 STEP 1 Graph functions Graph each function on a graphing calculator. a. b. STEP 2 Describe end behavior Summarize the end behavior of each function. Function As approaches 2` As approaches 1` a. f() 4 f() approaches 1` f() approaches 1` b. f() 2 4 f() approaches 2` f() approaches 2` DRAW CONCLUSIONS Use your observations to complete these eercises Graph the function. Then describe its end behavior as shown above. 1. f() 2. f() 2. f() 6 4. f() 2 6. Make a conjecture about the end behavior of each family of functions. a. f() n where n is odd b. f() 2 n where n is odd c. f() n where n is even d. f() 2 n where n is even 6. Make a conjecture about the end behavior of the function f() 6 2. Eplain your reasoning. 6 Chapter Polynomials and Polynomial Functions