Apply Exponent Properties Involving Quotients. Notice what happens when you divide powers with the same base. p a p a p a p a a

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1 8. Apply Eponent Properties Involving Quotients Before You used properties of eponents involving products. Now You will use properties of eponents involving quotients. Why? So you can compare magnitudes of earthquakes, as in E.. Key Vocabulary power, p. eponent, p. base, p. Notice what happens when you divide powers with the same base. a a p a p a p a p a a a p a p a a p a a a The eample above suggests the following property of eponents, known as the quotient of powers property. KEY CONCEPT For Your Notebook Quotient of Powers Property Let a be a nonzero real number, and let m and n be positive integers such that m > n. Words To divide powers having the same base, subtract eponents. Algebra am a n am n, a? 0 Eample E XAMPLE Use the quotient of powers property SIMPLIFY EXPRESSIONS When simplifying powers with numerical bases only, write your answers using eponents, as in parts (a), (b), and (c). a. c b. p 8 () 9 () ()9 8 () d. p GUIDED PRACTICE for Eample Simplify the epression... () 9 (). 9 p 9 9. y p y8 8. Apply Eponent Properties Involving Quotients 9

2 POWER OF A QUOTIENT Notice what happens when you raise a quotient to a power. a b a p a p a p a a p a p a p a a b b b b b p b p b p b b The eample above suggests the following property of eponents, known as the power of a quotient property. KEY CONCEPT For Your Notebook Power of a Quotient Property Let a and b be real numbers with b Þ 0, and let m be a positive integer. Words To find a power of a quotient, find the power of the numerator and the power of the denominator and divide. Algebra a b m am b m, b Þ 0 Eample SIMPLIFY EXPRESSIONS When simplifying powers with numerical and variable bases, evaluate the numerical power, as in part (b). E XAMPLE a. y y E XAMPLE a. y ( ) (y) Use the power of a quotient property Use properties of eponents Power of a quotient property b. () 9 p ( ) y Power of a product property y Power of a power property b. a b p (a ) p a b a Power of a quotient property a0 b p a Power of a power property a0 a b Multiply fractions. a8 b Quotient of powers property 9 Chapter 8 Eponents and Eponential Functions

3 GUIDED PRACTICE for Eamples and Simplify the epression.. a b. y. y 8. s t p t E XAMPLE Solve a multi-step problem FRACTAL TREE To construct what is known as a fractal tree, begin with a single segment (the trunk) that is unit long, as in Step 0. Add three shorter segments that are unit long to form the first set of branches, as in Step. Then continue adding sets of successively shorter branches so that each new set of branches is half the length of the previous set, as in Steps and. Step 0 Step Step Step a. Make a table showing the number of new branches at each step for Steps. Write the number of new branches as a power of. b. How many times greater is the number of new branches added at Step than the number of new branches added at Step? Solution a. Step Number of new branches 9 8 b. The number of new branches added at Step is. The number of new branches added at Step is. So, the number of new branches added at Step is times the number of new branches added at Step. GUIDED PRACTICE for Eample 9. FRACTAL TREE In Eample, add a column to the table for the length of the new branches at each step. Write the lengths of the new branches as powers of. What is the length of a new branch added at Step 9? 8. Apply Eponent Properties Involving Quotients 9

4 E XAMPLE Solve a real-world problem ASTRONOMY The luminosity (in watts) of a star is the total amount of energy emitted from the star per unit of time. The order of magnitude of the luminosity of the sun is 0 watts. The star Canopus is one of the brightest stars in the sky. The order of magnitude of the luminosity of Canopus is 0 0 watts. How many times more luminous is Canopus than the sun? Solution Luminosity of Canopus (watts) Luminosity of the sun (watts) Canopus c Canopus is about 0 times as luminous as the sun. GUIDED PRACTICE for Eample 0. WHAT IF? Sirius is considered the brightest star in the sky. Sirius is less luminous than Canopus, but Sirius appears to be brighter because it is much closer to Earth. The order of magnitude of the luminosity of Sirius is 0 8 watts. How many times more luminous is Canopus than Sirius? 8. EXERCISES SKILL PRACTICE HOMEWORK KEY WORKED-OUT SOLUTIONS on p. WS for Es. and STANDARDIZED TEST PRACTICE Es., 9,,, and MULTIPLE REPRESENTATIONS E. 9. VOCABULARY Copy and complete: In the power, is the? and is the?.. WRITING Eplain when and how to use the quotient of powers property. EXAMPLES and on pp. 99 for Es. 0 SIMPLIFYING EXPRESSIONS Simplify the epression. Write your answer using eponents () 8 (). () () 8. () 9 () 9. 0 p p p. p 9 9. p 8. 9 p 98 Chapter 8 Eponents and Eponential Functions

5 9. MULTIPLE CHOICE Which epression is equivalent to? A B C D 9 0. ERROR ANALYSIS Describe and correct the error in simplifying 9 p p EXAMPLES,, and on pp. 9 9 for Es. SIMPLIFYING EXPRESSIONS Simplify the epression.. y 8 p y. z 8 p z. a y 9. j k. p q.. 8. a b 9. c d 0. a b. y. y. y p. y p. p 8m m n. p y. MULTIPLE CHOICE Which epression is equivalent to y? A y B y 8 C 9 y D 9 y 8 SIMPLIFYING EXPRESSIONS Find the missing eponent. 8. (8) (8)? (8) 9.? p 0. p p p? p 9. c d? c d 8 SIMPLIFYING EXPRESSIONS Simplify the epression.. f g fg. s t st p (st) s t. m n m p mn n. y p y y. OPEN ENDED Write three epressions involving quotients that are equivalent to.. REASONING Name the definition or property that justifies each step to show that am a n a n m for m < n. Let m < n. Given a m a n am a n a m a m? a n a m? a n m? 8. CHALLENGE Find the values of and y if you know that b b y b9 and b p b b. Eplain how you found your answer. b y 8. Apply Eponent Properties Involving Quotients 99

6 PROBLEM SOLVING EXAMPLES and on pp. 998 for Es MULTIPLE REPRESENTATIONS Draw a square with side lengths that are unit long. Divide it into four new squares with side lengths that are one half the side length of the original square, as shown in Step. Keep dividing the squares into new squares, as shown in Steps and. Step 0 Step Step Step a. Making a Table Make a table showing the number of new squares and the side length of a new square at each step for Steps. Write the number of new squares as a power of. Write the side length of a new square as a power of. b. Writing an Epression Write and simplify an epression to find by how many times the number of squares increased from Step to Step. 0. GROSS DOMESTIC PRODUCT In 00 the gross domestic product (GDP) for the United States was about trillion dollars, and the order of magnitude of the population of the U.S. was 0 8. Use order of magnitude to find the approimate per capita (per person) GDP?. SPACE TRAVEL Alpha Centauri is the closest star system to Earth. Alpha Centauri is about 0 kilometers away from Earth. A spacecraft leaves Earth and travels at an average speed of 0 meters per second. About how many years would it take the spacecraft to reach Alpha Centauri?. ASTRONOMY The brightness of one star relative to another star can be measured by comparing the magnitudes of the stars. For every increase in magnitude of, the relative brightness is diminished by a factor of.. For instance, a star of magnitude 8 is. times less bright than a star of magnitude. The constellation Ursa Minor (the Little Dipper) is shown. How many times less bright is Eta Ursae Minoris than Polaris? Ursa Minor Eta Ursae Minoris (magnitude ) Polaris (magnitude ). EARTHQUAKES The energy released by one earthquake relative to another earthquake can be measured by comparing the magnitudes (as determined by the Richter scale) of the earthquakes. For every increase of in magnitude, the energy released is multiplied by a factor of about. How many times greater is the energy released by an earthquake of magnitude than the energy released by an earthquake of magnitude? WORKED-OUT SOLUTIONS 00 Chapter 8 Eponents p. WSand Eponential Functions STANDARDIZED TEST PRACTICE MULTIPLE REPRESENTATIONS

7 . EXTENDED RESPONSE A byte is a unit used to measure computer memory. Other units are based on the number of bytes they represent. The table shows the number of bytes in certain units. For eample, from the table you can calculate that terabyte is equivalent to 0 gigabytes. a. Calculate How many kilobytes are there in terabyte? Unit Number of bytes b. Calculate How many megabytes are there in Kilobyte 0 petabyte? Megabyte 0 c. CHALLENGE Another unit used to measure computer memory is a bit. There are 8 bits Gigabyte 0 in a byte. Eplain how you can convert the Terabyte 0 number of bytes per unit given in the table to the number of bits per unit. Petabyte 0 MIXED REVIEW PREVIEW Prepare for Lesson 8. in Es. 0. Solve the equation. Check your solution. (p. ). k 9. t. v 8. y 9. z 0. z Write an equation of the line that passes through the given points. (p. 9). (, ), (0, ). (0, ), (, ). (0, ), (, ). (, ), (, ). (, ), (, ). (, ), (, ) QUIZ for Lessons Simplify the epression. Write your answer using eponents.. p (p. 89). ( ) (p. 89). ( p ) (p. 89). p p (p. 89). ()() 9 (p. 89).. (p. 9) (9) 9 (9) (p. 9) 8. p (p. 9) 9. (p. 9) Simplify the epression. 0. p (p. 89). ( ) (p. 89). () (p. 89). ( ) p (p. 89). ( ) ( ) (p. 89). 9 p (p. 9). (p. 9). w v (p. 9) 8. (p. 9) 9. MAPLE SYRUP PRODUCTION In 00 the order of magnitude of the number of maple syrup taps in Vermont was 0. The order of magnitude of the number of gallons of maple syrup produced in Vermont was 0. About how many gallons of syrup were produced per tap in Vermont in 00? (p. 9) EXTRA PRACTICE for Lesson 8., p. 9 ONLINE QUIZ at classzone.com 0

8 Investigating g Algebra ACTIVITY Use before Lesson Zero and Negative Eponents MATERIALS paper and pencil QUESTION E XPLORE How can you simplify epressions with zero or negative eponents? Evaluate powers with zero and negative eponents STEP Find a pattern Copy and complete the tables for the powers of and. Eponent, n Value of n??? Eponent, n Value of n 8??? As you read the tables from the bottom up, you see that each time the eponent is increased by, the value of the power is multiplied by the base. What can you say about the eponents and the values of the powers as you read the table from the top down? STEP Etend the pattern Copy and complete the tables using the pattern you observed in Step. Eponent, n Power, n Eponent, n Power, n 8???? 0? 0????? DRAW CONCLUSIONS Use your observations to complete these eercises. Find n and n for n,, and.. What appears to be the value of a 0 for any nonzero number a?. Write each power in the tables above as a power with a positive eponent. For eample, you can write as. 0 Chapter 8 Eponents and Eponential Functions