Page of 6 6. Using Properties of Eponents What you should learn GOAL Use properties of eponents to evaluate and simplify epressions involving powers. GOAL Use eponents and scientific notation to solve real-life problems, such as finding the per capita GDP of Denmark in Eample. Why you should learn it To simplify real-life epressions, such as the ratio of a state s park space to total area in E. 7. GOAL PROPERTIES OF EXPONENTS Recall that the epression a n, where n is a positive integer, represents the product that you obtain when a is used as a factor n times. In the activity you will investigate two properties of eponents. ACTIVITY Developing Concepts Products and Quotients of Powers How many factors of are there in the product? Use your answer to write the product as a single power of. Write each product as a single power of by counting the factors of. Use a calculator to check your answers. a. b. 6 c. 6 d. Complete this equation: m n =? Write each quotient as a single power of by first writing the numerator and denominator in epanded form (for eample, = ) and then canceling common factors. Use a calculator to check your answers. a. b. c. 7 d. 6 Complete this equation: m =? n In the activity you may have discovered two of the following properties of eponents. Lake Clark National Park, Alaska CONCEPT SUMMARY Let a and b be real numbers and let m and n be integers. PRODUCT OF POWERS PROPERTY POWER OF A POWER PROPERTY POWER OF A PRODUCT PROPERTY PROPERTIES OF EXPONENTS a m a n = a m + n (a m ) n = a mn (ab) m = a m b m NEGATIVE EXPONENT PROPERTY a ºm =, a 0 a m ZERO EXPONENT PROPERTY a 0 =, a 0 a QUOTIENT OF POWERS PROPERTY m = a m º n, a 0 a n a POWER OF A QUOTIENT PROPERTY b m a = m, b 0 b m 6. Using Properties of Eponents
Page of 6 The properties of eponents can be used to evaluate numerical epressions and to simplify algebraic epressions. In this book we assume that any base with a zero or negative eponent is nonzero. A simplified algebraic epression contains only positive eponents. E X A M P L E Evaluating Numerical Epressions a. ( ) = Power of a power property = Simplify eponent. Study Tip When you multiply powers, do not multiply the bases. For eample, 8. = 096 Evaluate power. b. = Power of a quotient property = Evaluate powers. 96 c. (º) º6 (º) = (º) º6 + Product of powers property = (º) º Simplify eponent. = (º ) = Negative eponent property Evaluate power. E X A M P L E Simplifying Algebraic Epressions r a. s º = r (s º ) Power of a quotient property r = Power of a power property s º0 = r s 0 Negative eponent property HOMEWORK HELP Visit our Web site for etra eamples. b. (7b º ) b b = 7 (b º ) b b Power of a product property = 9b º6 b b Power of a power property = 9b º6 + + Product of powers property = 9b 0 Simplify eponent. = 9 Zero eponent property (y ) (y ) c. = y º y º Power of a product property = y y º Power of a power property = º y º (º) Quotient of powers property = º y Simplify eponents. = y Negative eponent property Chapter 6 Polynomials and Polynomial Functions
Page of 6 FOCUS ON APPLICATIONS GOAL USING PROPERTIES OF EXPONENTS IN Earth E X A M P L E Comparing Real-Life Volumes Jupiter ASTRONOMY The radius of the sun is about 09 times as great as Earth s radius. How many times as great as Earth s volume is the sun s volume? ASTRONOMY Jupiter is the largest planet in the solar system. It has a radius of 7,00 km over times as great as Earth s, but only about one tenth as great as the sun s. APPLICATION LINK Sun SOLUTION Let r represent Earth s radius. Sun s volume π(09r) = The volume of a sphere is } }πr. Earth s volume πr π09 r = Power of a product property πr = 09 r 0 Quotient of powers property = 09 Zero eponent property =,9,09 Evaluate power. The sun s volume is about. million times as great as Earth s volume........... Skills Review For help with scientific notation, see p. 9. A number is epressed in scientific notation if it is in the form c ª 0 n where c < 0 and n is an integer. For instance, the width of a molecule of water is about. ª 0 º8 meter, or 0.0000000 meter. When working with numbers in scientific notation, the properties of eponents listed on page can help make calculations easier. Economics E X A M P L E Using Scientific Notation in Real Life In 997 Denmark had a population of,8,000 and a gross domestic product (GDP) of $,00,000,000. Estimate the per capita GDP of Denmark. DATA UPDATE of UN/ECE Statistical Division data at SOLUTION Per capita means per person, so divide the GDP by the population. GDP =,00,000,000 Divide GDP by population. Pop ulation,8,000. ª 0 = Write in scientific notation..8 ª 0 6 =. ª 0. 8 0.9 ª 0 Quotient of powers property Use a calculator. =,900 Write in standard notation. The per capita GDP of Denmark in 997 was about $,000 per person. 6. Using Properties of Eponents
Page of 6 GUIDED PRACTICE Vocabulary Check. State the name of the property illustrated. Concept Check a. a m a n = a m + n b. (a m ) n = a mn c. (ab) m = a m b m. ERROR ANALYSIS Describe the mistake made in simplifying the epression. 8 a. (º) (º) = b. c. = = Skill Check Evaluate the epression. Tell which properties of eponents you used.. 6 6. (9 6 )(9 ) º. ( ) 6. 7. º 8. 7 7 º º º Simplify the epression. Tell which properties of eponents you used. 9. z º z º z 6 0. yz º ( y) z. ( ) º. 6 y 6 (y).. º y y º. ASTRONOMY Earth has a radius of about 6.8 ª 0 kilometers. The sun has a radius of about 6.96 ª 0 kilometers. Use the formula for the volume of a sphere given on page to calculate the volume of the sun and the volume of Earth. Divide the volumes. Do you get the same result as in Eample? PRACTICE AND APPLICATIONS Etra Practice to help you master skills is on p. 97. EVALUATING NUMERICAL EXPRESSIONS Evaluate the epression. Tell which properties of eponents you used. 6. 7. ( º ) 8. (º9)(º9) 9. (8 ) 0.. 7. 9 º. º 0 º º.. 8 º 6. ( º ) º 7. 6 8. 9. 6 0 6 6 º 0. 0 (6 0 º ) º º. º SIMPLIFYING ALGEBRAIC EXPRESSIONS Simplify the epression. Tell which properties of eponents you used.. 8. ( ). ( y ) º. º9 º HOMEWORK HELP Eample : Es. 6 Eample : Es. Eamples, : Es. 6 y 6. 7. ( y 7 ) º y 0 8. 9. º º y 0 y 0 º y º º y y º 0. (0 y ) º.. ( y ) º. y 6y º. y 9 y. 9 º7y y 6. 0 0 y 7. 6 7 y 0 y º y 7 y º y 6 Chapter 6 Polynomials and Polynomial Functions
Page of 6 GEOMETRY CONNECTION Write an epression for the area or volume of the figure in terms of. 8. A = s 9. A = πr 0. V = πr h. V = πr SCIENTIFIC NOTATION In Eercises 6, use scientific notation.. NATIONAL DEBT On June 8, 999, the national debt of the United States was about $,608,000,000,000. The population of the United States at that time was about 7,000,000. Suppose the national debt was divided evenly among everyone in the United States. How much would each person owe? DATA UPDATE of Bureau of the Public Debt and U.S. Census Bureau data at. SOCIAL STUDIES CONNECTION The table shows the population and gross domestic product (GDP) in 997 for each of si different countries. Calculate the per capita GDP for each country. DATA UPDATE of UN/ECE Statistical Division data at FOCUS ON CAREERS Country Population GDP (U.S. dollars) France 8,607,000,9,600,000,000 Germany 8,06,000,89,00,000,000 Ireland,66,000 7,00,000,000 Luembourg 0,000,600,000,000 The Netherlands,600,000,00,000,000 Sweden 8,89,000 77,00,000,000. BIOLOGY CONNECTION A red blood cell has a diameter of approimately 0.0007 centimeter. Suppose one of the arteries in your body has a diameter of 0.06 centimeter. How many red blood cells could fit across the artery? ORNITHOLOGIST An ornithologist is a scientist who studies the history, classification, biology, and behavior of birds. CAREER LINK. SPACE EXPLORATION On February 7, 998, Voyager became the most distant manmade object in space, at a distance of 0,00,000,000 kilometers from Earth. How long did it take Voyager to travel this distance given that it traveled an average of,90,000 kilometers per day? Source: NASA 6. ORNITHOLOGY Some scientists estimate that there are about 8600 species of birds in the world. The mean number of birds per species is approimately,000,000. About how many birds are there in the world? 6. Using Properties of Eponents 7
Page 6 of 6 Test Preparation 7. MULTI-STEP PROBLEM Suppose you live in a state that has a total area of.8 ª 0 7 acres and.9 ª 0 acres of park space. You think that the state should set aside more land for parks. The table shows the total area and the amount of park space for several states. State Total area (acres) Amount of park space (acres) Alaska 9,77,00,0,000 California 0,676,000,,000 Connecticut,8,000 76,000 Kansas,660,000 9,000 Ohio 8,690,000 0,000 Pennsylvania 9,77,000 8,000 Challenge EXTRA CHALLENGE Source: Statistical Abstract of the United States a. Write the total area and the amount of park space for each state in scientific notation. b. For each state, divide the amount of park space by the total area. c. Writing You want to ask the state legislature to increase the amount of park space in your state. Use your results from parts (a) and (b) to write a letter that eplains why your state needs more park space. LOGICAL REASONING In Eercises 8 and 9, refer to the properties of eponents on page. 8. Show how the negative eponent property can be derived from the quotient of powers property and the zero eponent property. 9. Show how the quotient of powers property can be derived from the product of powers property and the negative eponent property. MIXED REVIEW GRAPHING Graph the equation. (Review.,. for 6.) 60. y = º 6. y = º º 6. y = + 6. y = º + 6. y = + 6. y = º( + 6) 66. y = º º 6 67. y = º + 0 68. y = º( º ) + 8 SOLVING QUADRATIC EQUATIONS Solve the equation. (Review.) 69. = 70. º = º 7. = 6 7. º 8 = 00 7. º = 8 7. º = 9 7. º + 9 = º 6 76. + = º 8 77. º + 7 = º OPERATIONS WITH COMPLEX NUMBERS Write the epression as a comple number in standard form. (Review.) 78. (9 + i) + (9 º i) 79. (º + i) º (º º i) 80. (0 º i) º ( + 7i) 8. ºi(7 + i) 8. ºi( + i) 8. ( + i)(9 + i) 8 Chapter 6 Polynomials and Polynomial Functions