Properties of Rational Exponents PROPERTIES OF RATIONAL EXPONENTS AND RADICALS. =, a 0 25 º1/ =, b /3 2. b m

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1 Page of 8. Properties of Rational Eponents What ou should learn GOAL Use properties of rational eponents to evaluate and simplif epressions. GOAL Use properties of rational eponents to solve real-life problems, such as finding the surface area of a mammal in Eample 8. Wh ou should learn it To model real-life quantities, such as the frequencies in the musical range of a trumpet for E. 9. REAL LIFE GOAL PROPERTIES OF RATIONAL EXPONENTS AND RADICALS The properties of integer eponents presented in Lesson 6. can also be applied to rational eponents. CONCEPT SUMMARY Let a and b be real numbers and let m and n be rational numbers. The following properties have the same names as those listed on page, but now appl to rational eponents as illustrated. PROPERTY PROPERTIES OF RATIONAL EXPONENTS EXAMPLE. a m a n = a m + n / / = (/ + /) = = 9. (a m ) n = a mn ( / ) = (/ ) = = 6. (ab) m = a m b m (9 ) / = 9 / / = = 6. a ºm =, a 0 5 º/ = = a m 5 / 5 a 5. m = a m º n, a 0 6 = 6 (5/ º /) = 6 = 6 a n / 6 5/ a 6. b m a 8 =, b 0 m = = 8 b m If m = n for some integer n greater than, the third and sith properties can be written using radical notation as follows: n a b = n a n b n a = b n a n b Product propert Quotient propert EXAMPLE Using Properties of Rational Eponents Look Back For help with properties of eponents, see p.. Use the properties of rational eponents to simplif the epression. a. 5 / 5 / = 5 (/ + /) = 5 / b. (8 / 5 ) = (8 / ) (5 ) = 8 (/ ) 5 ( ) = 8 5 / = 8 5 / c. ( ) º/ = ( ) º/ = (6 ) º/ = 6 [ (º/)] = 6 º = 6 d. = = ( º ) = / e. = = ( ) = ( ) = /. Properties of Rational Eponents 0

2 Page of 8 EXAMPLE Using Properties of Radicals Use the properties of radicals to simplif the epression. a. 6 = 6 = 6 = Use the product propert. 6 b. = = 8 = Use the quotient propert For a radical to be in simplest form, ou must not onl appl the properties of radicals, but also remove an perfect nth powers (other than ) and rationalize an denominators. EXAMPLE Writing Radicals in Simplest Form HOMEWORK HELP Visit our Web site for etra eamples. INTERNET Write the epression in simplest form. a. 5 = Factor out perfect cube. = Product propert = Simplif. b. 5 = Make the denominator a perfect fifth power = 5 Simplif. 5 = Quotient propert 5 5 = Simplif. Two radical epressions are like radicals if the have the same inde and the same radicand. For instance, and are like radicals. To add or subtract like radicals, use the distributive propert. EXAMPLE Adding and Subtracting Roots and Radicals Perform the indicated operation. a. (6 /5 ) + (6 /5 ) = ( + )(6 /5 ) = 9(6 /5 ) b. 6 º = 8 º = 8 º = º = ( º ) = 08 Chapter Powers, Roots, and Radicals

3 Page of 8 The properties of rational eponents and radicals can also be applied to epressions involving variables. Because a variable can be positive, negative or zero, sometimes absolute value is needed when simplifing a variable epression. n n = when n is odd = and (º) = º n n = when n is even 5 = 5 and (º5) = 5 Absolute value is not needed when all variables are assumed to be positive. EXAMPLE 5 Simplifing Epressions Involving Variables Simplif the epression. Assume all variables are positive. a. 5 6 = 5( ) = 5 b. (9u v 0 ) / = 9 / (u ) / (v 0 ) / = u ( /) v (0 /) = uv 5 c. = = = 8 8 ( ) / 6 d. = ( º ) / z º(º5) = / / z 5 / º 5 z EXAMPLE 6 Writing Variable Epressions in Simplest Form HOMEWORK HELP Visit our Web site for etra eamples. INTERNET Write the epression in simplest form. Assume all variables are positive. a. 5 5a 5 b 9 c = 5 5a 5 b 5 b c 0 c Factor out perfect fifth powers. = 5 a 5 b 5 c 0 5 5b c Product propert = abc 5 5b c Simplif. b. = Make the denominator a perfect cube. = Simplif. 9 9 = Quotient propert = Simplif. EXAMPLE Adding and Subtracting Epressions Involving Variables Perform the indicated operation. Assume all variables are positive. a = (5 + 6) = b. º = ( º ) = º5 c. 5 5 º 0 = 5 º 5 = ( º ) 5 = 5. Properties of Rational Eponents 09

Page of 8 GOAL PROPERTIES OF RATIONAL EXPONENTS IN REAL LIFE REAL LIFE EXAMPLE 8 Evaluating a Model Using Properties of Rational Eponents Biolog Biologists stud characteristics of various living

4 Page of 8 GOAL PROPERTIES OF RATIONAL EXPONENTS IN REAL LIFE REAL LIFE EXAMPLE 8 Evaluating a Model Using Properties of Rational Eponents Biolog Biologists stud characteristics of various living things. One wa of comparing different animals is to compare their sizes. For eample, a mammal s surface area S (in square centimeters) can be approimated b the model S = km / where m is the mass (in grams) of the mammal and k is a constant. The values of k for several mammals are given in the table. Mammal Mouse Cat Large dog Cow Rabbit Human k Approimate the surface area of a cat that has a mass of 5 kilograms (5 ª 0 grams). Source: Scaling: Wh Is Animal Size So Important? SOLUTION S = km / Write model. = 0.0(5 ª 0 ) / Substitute for k and m. = 0.0(5) / (0 ) / Power of a product propert 0.0(.9)(0 ) Power of a power propert = 90 Simplif. The cat s surface area is approimatel 000 square centimeters. EXAMPLE 9 Using Properties of Rational Eponents with Variables BIOLOGY CONNECTION You are studing a Canadian ln whose mass is twice the mass of an average house cat. Is its surface area also twice that of an average house cat? BIOLOGY The average mass of a Canadian ln is about 9. kilograms. The average mass of a house cat is about.8 kilograms. REAL FOCUS ON APPLICATIONS LIFE SOLUTION Let m be the mass of an average house cat. Then the mass of the Canadian ln is m. The surface areas of the house cat and the Canadian ln can be approimated b: S cat = 0.0m / S ln = 0.0(m) / To compare the surface areas look at their ratio. S ln 0.0(m) = / Write ratio of surface areas. Scat 0.0m / 0.0( / )(m / ) = Power of a product propert 0.0m / = / Simplif..59 Evaluate. The surface area of the Canadian ln is about one and a half times that of an average house cat, not twice as much. 0 Chapter Powers, Roots, and Radicals

5 Page 5 of 8 GUIDED PRACTICE Vocabular Check Concept Check. List three pairs of like radicals.. If ou know that 6,656,000 = 9 6 5, what is the cube root of 6,656,000? Eplain our reasoning. ERROR ANALYSIS Eplain the error made in simplifing the epression = 5 0. = = 8 ( 8 ) Skill Check Simplif the epression. 5. / / 6. (5 ) º/ / + (8 / ). 00 º Simplif the epression. Assume all variables are positive.. / /. ( /6 ) 5. a 6 6. b º 0 5 z a /5 º 6a / BIOLOGY CONNECTION The average mass of a rabbit is.6 kilograms. Use the information given in Eample 8 to approimate the surface area of a rabbit. PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p. 99. PROPERTIES OF RATIONAL EXPONENTS Simplif the epression.. 5/. (5 / ) /. / 6 / ( / ) 9. / º/ /5 6 º/ / 5. /9 5 / /9. 0/8. (0 / / ) º / º/8 5 / PROPERTIES OF RADICALS Simplif the epression ( 6 6) HOMEWORK HELP Eample : Es. Eample : Es. Eample : Es. 9 Eample : Es Eample 5: Es Eample 6: Es Eample : Es. 6 8 Eample 8: Es Eample 9: Es SIMPLEST FORM Write the epression in simplest form COMBINING ROOTS AND RADICALS Perform the indicated operation (5) / º (5) / 5. º / º 0 / Properties of Rational Eponents

6 Page 6 of 8 VARIABLE EXPRESSIONS Simplif the epression. Assume all variables are positive. 56. /5 5. ( ) / º5/ / / ( / ) / 6. ( 5 ) º SIMPLEST FORM Write the epression in simplest form. Assume all variables are positive z z z COMBINING VARIABLE EXPRESSIONS Perform the indicated operation. Assume all variables are positive /5 º /5 8. º + 9. ( 9 ) + ( /9 ) º º EXTENSION: IRRATIONAL EXPONENTS The properties ou studied in this lesson can also be applied to irrational eponents. Simplif the epression. Assume all variables are positive ( ) 8. () π 85. º 5 / z º / z / /π º/ z π º /π 5 Skills Review For help with perimeter and area, see p GEOMETRY CONNECTION Find a radical epression for the perimeter of the triangle. Simplif the epression. 9. GEOMETRY CONNECTION The areas of two circles are 5 square centimeters and 0 square centimeters. Find the eact ratio of the radius of the smaller circle to the radius of the larger circle. 9. BIOLOGY CONNECTION Look back at Eample 8. Approimate the surface area of a human that has a mass of 68 kilograms. 9. PINHOLE CAMERA A pinhole camera is made out of a light-tight bo with a piece of film attached to one side and a pinhole on the opposite side. The optimum diameter d (in millimeters) of the pinhole can be modeled b d =.9(5.5 ª 0 º )l /, where l is the length of the camera bo (in millimeters). What is the optimum diameter for a pinhole camera if the camera bo has a length of 0 centimeters? E. 90 pinhole tree film l Chapter Powers, Roots, and Radicals

7 Page of 8 MUSIC In Eercises 9 and 95, use the following information. The musical note A-0 (the A above middle C) has a frequenc of 0 vibrations per second. The frequenc ƒ of an note can be found using ƒ = 0 n/ where n represents the number of black and white kes the given note is above or below A-0. For notes above A-0, n > 0, and for notes below A-0, n < 0. trumpet middle C A-0 ABCDE F GABC D E FGABC D E FGABC D E FGABC D E FGABC D E FGABC D E FGABC Test Preparation Challenge 9. Find the highest and lowest frequencies in the musical range of a trumpet. What is the eact ratio of these two frequencies? 95. LOGICAL REASONING Describe the pattern of the frequencies of successive notes with the same letter. 96. DISTANCE OF AN OBJECT The maimum horizontal distance d that an object can travel when launched at an optimum angle of projection from an v 0 (v 0 ) + gh 0 initial height h 0 can be modeled b d = where v 0 is the initial g speed and g is the acceleration due to gravit. Simplif the model when h 0 =0. 9. BALLOONS You have filled two round balloons with air. One balloon has twice as much air as the other balloon. The formula for the surface area S of a sphere in terms of its volume V is S = (π) (V) /. B what factor is the surface area of the larger balloon greater than that of the smaller balloon? 98. MULTI-STEP PROBLEM A common ant absorbs ogen at a rate of about 6. milliliters per second per square centimeter of eoskeleton. It needs about milliliters of ogen per second per cubic centimeter of its bod. An ant is basicall clindrical in shape, so its surface area S and volume V can be approimated b the formulas for the surface area and volume of a clinder: S = πrh + πr V = πr h a. Approimate the surface area and volume of an ant that is 8 millimeters long and has a radius of.5 millimeters. Would this ant have a surface area large enough to meet its ogen needs? b. Consider a giant ant that is 8 meters long and has a radius of.5 meters. Would this ant have a surface area large enough to meet its ogen needs? c. Writing Substitute 000r for r and 000h for h into the formulas for surface area and volume. How does increasing the radius and height b a factor of 000 affect surface area? How does it affect volume? Use the results to eplain wh giant ants do not eist. 99. CRITICAL THINKING Substitute different combinations of odd and even positive integers for m and n in the epression n m. Do not assume that is alwas positive. When is absolute value needed in simplifing the epression?. Properties of Rational Eponents

8 Page 8 of 8 MIXED REVIEW COMPLETING THE SQUARE Find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a binomial. (Review 5.5) c 0. º + c 0. º.6 + c c c 05. º + c POLYNOMIAL OPERATIONS Perform the indicated operation. (Review 6. for.) 06. (º + 6) º (8 + º ) 0. (50 º ) + ( ) ( º 9) 09. ( + ) LONG DIVISION Divide using long division. (Review 6.5 for.) 0. ( º 8 º 8) ( + ). ( + º ) ( + ). ( º 6) ( º ). ( º º 0 + 0) ( º 5) QUIZ Self-Test for Lessons. and. Evaluate the epression without using a calculator. (Lesson.). 8 /. º/5. º(8 / ). (º6) / Solve the equation. Round our answer to two decimal places. (Lesson.) 5. 5 = 0 6. º9 6 = 8. º = 9 8. ( + ) = º5 Write the epression in simplest form. (Lesson.) º/ /5 + (8 /5 ) Write the epression in simplest form. Assume all variables are positive. (Lesson.) ( /5 ) 5/ / / º (9) / º ( ) /. GENERATING POWER As a rule of thumb, the power P (in horsepower) d that a ship needs can be modeled b P = / s where d is the ship s c displacement (in tons), s is the normal speed (in knots), and c is the Admiralt coefficient. If a ship displaces 0,090 tons, has a normal speed of.5 knots, and has an Admiralt coefficient of 0, how much power does it need? (Lesson.). BIOLOGY CONNECTION The surface area S (in square centimeters) of a large dog can be approimated b the model S =.m / where m is the mass (in grams) of the dog. A Labrador retriever s mass is about three times the mass of a Scottish terrier. Is its surface area also three times that of a Scottish terrier? (Lesson.) Chapter Powers, Roots, and Radicals

Evaluate nth Roots and Use Rational Exponents. p Evaluate nth roots and study rational exponents. VOCABULARY. Index of a radical

. Georgia Performance Standard(s) MMA2a, MMA2b, MMAd Your Notes Evaluate nth Roots and Use Rational Eponents Goal VOCABULARY nth root of a p Evaluate nth roots and stud rational eponents. Inde of a radical