4 Rational Exponents And Radical Functions
44.14.24.34.44.54.6Rational Exponents andRadical Functionsnth Roots and Rational ExponentsProperties of Rational Exponents and RadicalsGraphing Radical FunctionsSolving Radical Equations and InequalitiesPerforming Function OperationsInverse of a FunctionSEE the Big IdeaHull Speed (p. 238)White Rhino (p. 228)Concert (p. 224)Mars Rover (p. 210)Constellations (p. 206)Int Math3 PE 04OP.indd 1901/30/15 3:31 PM
Maintaining Mathematical ProficiencyProperties of Integer ExponentsExample 1 x5 x2Simplify the expression —.x3 x5 x2xx5 2x37x —3x x7 3 x4— —3Example 2Product of Powers PropertyAdd exponents.Quotient of Powers PropertySubtract exponents.( )2s3 2Simplify the expression — .t( )2s3t—2(2s3)2 —t222 (s3)2 —t24s6 —t2Power of a Quotient Property Power of a Product PropertyPower of a Power PropertySimplify the expression. 1. y6 yx6xn4n3. —62x5( )6.2. —3 4. —5 3x25. xm m(—z m )x4w3 3—22z2723Rewriting Literal EquationsExample 3Solve the literal equation 5y 2x 10 for y. 5y 2x 10Write the equation. 5y 2x 2x 10 2xAdd 2x to each side. 5y 10 2xSimplify. 5y 5Divide each side by 5.10 2x 52y 2 —x5— —Simplify.Solve the literal equation for y.7. 4x y 210. 2xy 6y 108. x —3 y 119. 2y 9 13x11. 8x 4xy 312. 6x 7xy 1513. ABSTRACT REASONING Is the order in which you apply properties of exponents important?Explain your reasoning.Dynamic Solutions available at BigIdeasMath.comInt Math3 PE 04OP.indd 1911911/30/15 3:31 PM
MathematicalPracticesMathematically proficient students express numerical answers precisely.Using Technology to Evaluate RootsCore ConceptEvaluating Roots with a CalculatorExamplesquare root— 64 8Square root: (64)cube root3— 64 4Cube root:fourth root4—Fourth root: 256 4Fifth root: 32 25—83 (64)44x (256)45x (32)2fifth rootApproximating RootsEvaluate each root using a calculator. Round your answer to two decimal places.a.— 50b.3— 50c.4—5— 50d. 50SOLUTIONa.b.c.— 50 7.073— 50 3.684— 50 2.665—d. 50 2.19Round down.Round down.Round up. (50)3 (50)4x (50)5x und up.Monitoring Progress1. Use the Pythagorean Theorem to find1 in.the exact lengths of a, b, c, and din the figure.1 in.2. Use a calculator to approximate eachlength to the nearest tenth of an inch inMonitoring Progress Question 1.b3. Use a ruler to check the reasonablenessof your answers in Monitoring ProgressQuestion 2.1 in.cda1 in.1 in.192Chapter 4Int Math3 PE 04OP.indd 192Rational Exponents and Radical Functions1/30/15 3:31 PM
4.1nth Roots and Rational ExponentsEssential QuestionHow can you use a rational exponent torepresent a power involving a radical?Previously, you learned that the nth root of a can be represented asn— a a1/nDefinition of rational exponentfor any real number a and integer n greater than 1.Exploring the Definition of aRational ExponentCONSTRUCTINGVIABLE ARGUMENTSTo be proficient in math,you need to understandand use stated definitionsand previously establishedresults.Work with a partner. Use a calculator to show that each statement is true.——a. 9 91/23—c. 8 81/3b. 2 21/24—3—4—e. 16 161/4d. 3 31/3f. 12 121/4Writing Expressions inRational Exponent FormWork with a partner. Use the definition of a rational exponent and the properties ofexponents to write each expression as a base with a single rational exponent. Then usea calculator to evaluate each expression. Round your answer to two decimal places.Sample4 (2/3)( 3 —4 )2 (41/3)22.5198421 42/3 2.52— 3a. ( 5 )5— 4d. ( 10 )4— 2b. ( 4 )— 3e. ( 15 )3— 2c. ( 9 )3— 4f. ( 27 )Writing Expressions in Radical FormWork with a partner. Use the properties of exponents and the definition of arational exponent to write each expression as a radical raised to an exponent. Then usea calculator to evaluate each expression. Round your answer to two decimal places.3— 2Sample 52/3 (51/3)2 ( 5 ) 2.92a. 82/3b. 65/2c. 123/4d. 103/2e. 163/2f. 206/5Communicate Your Answer4. How can you use a rational exponent to represent a power involving a radical?5. Evaluate each expression without using a calculator. Explain your reasoning.a. 43/2d. 493/2b. 324/5e. 1254/3Section 4.1Int Math3 PE 0401.indd 193c. 6253/4f. 1006/3nth Roots and Rational Exponents1931/30/15 3:32 PM
4.1LessonWhat You Will LearnFind nth roots of numbers.Evaluate expressions with rational exponents.Core VocabulVocabularylarrynth root of a, p. 194index of a radical, p. 194Previoussquare rootcube rootexponentSolve equations using nth roots.nth RootsYou can extend the concept of a square root to other types of roots. For example, 2 isa cube root of 8 because 23 8. In general, for an integer n greater than 1, if b n a,n—then b is an nth root of a. An nth root of a is written as a , where n is the index ofthe radical.You can also write an nth root of a as a power of a. If you assume the Power of aPower Property applies to rational exponents, then the following is true.(a1/2)2 a(1/2) 2 a1 a(a1/3)3 a(1/3) 3 a1 a(a1/4)4 a(1/4) 4 a1 a—Because a1/2 is a number whose square is a, you can write a a1/2. Similarly,3—n—4— a a1/3 and a a1/4. In general, a a1/n for any integer n greater than 1.UNDERSTANDINGMATHEMATICALTERMSWhen n is even and a 0,there are two real roots.The positive root is calledthe principal root.Core ConceptReal nth Roots of aLet n be an integer (n 1) and let a be a real number.n is an even integer.n is an odd integer.—a 0 No real nth rootsna 0 One real nth root: a a1/nn—n—a 0 One real nth root: 0 0a 0 One real nth root: 0 0—na 0 Two real nth roots: a a1/n—na 0 One real nth root: a a1/nFinding nth RootsFind the indicated real nth root(s) of a.a. n 3, a 216b. n 4, a 81SOLUTIONa. Because n 3 is odd and a 216 0, 216 has one real cube root.3—Because ( 6)3 216, you can write 216 6 or ( 216)1/3 6.b. Because n 4 is even and a 81 0, 81 has two real fourth roots.4—Because 34 81 and ( 3)4 81, you can write 81 3 or 811/4 3.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind the indicated real nth root(s) of a.194Chapter 4Int Math3 PE 0401.indd 1941. n 4, a 162. n 2, a 493. n 3, a 1254. n 5, a 243Rational Exponents and Radical Functions1/30/15 3:32 PM
Rational ExponentsA rational exponent does not have to be of the form 1/n. Other rational numbers, suchas 3/2 and 1/2, can also be used as exponents. Two properties of rational exponentsare shown below.Core ConceptRational ExponentsLet a1/n be an nth root of a, and let m be a positive integer.— mnam/n (a1/n)m ( a)111,a 0a m/n — — —n— mam/n (a1/n)m ( a)Evaluating Expressions with Rational ExponentsEvaluate each expression.b. 32 3/5a. 163/2SOLUTIONRadical FormRational Exponent Forma. 163/2 (161/2)3 43 641111b. 32 3/5 — — — —323/5 (321/5)3 23 8COMMON ERRORBe sure to use parenthesesto enclose a rationalexponent: 9 (1/5) 1.55.Without them, thecalculator evaluates apower and then divides:9 1/5 1.8.— 3163/2 ( 16 ) 43 64111132 3/5 — — — —323/5 ( 5 — )3 23 8 32When using a calculator to approximate an nth root, you may want to rewrite the nthroot in rational exponent form.Approximating Expressions with RationalExponentsEvaluate each expression using a calculator. Round your answer to two decimal places.a. 91/54— 3c. ( 7 )b. 123/8SOLUTIONa. 91/5 1.55b. 123/8 2.549 (1/5)12 (3/8)4— 3c. Before evaluating ( 7 ) , rewrite theexpression in rational exponent form.7 (3/4)1.5518455742.5391769514.303517071( 4 —7 )3 73/4 4.30Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comEvaluate the expression without using a calculator.5. 45/26. 9 1/27. 813/48. 17/8Evaluate the expression using a calculator. Round your answer to two decimalplaces when appropriate.9. 62/510. 64 2/3Section 4.1Int Math3 PE 0401.indd 1954— 511. ( 16 )3— 212. ( 30 )nth Roots and Rational Exponents1951/30/15 3:32 PM
Solving Equations Using nth RootsTo solve an equation of the form u n d, where u is an algebraic expression, take thenth root of each side.Solving Equations Using nth RootsFind the real solution(s) of (a) 4x5 128 and (b) (x 3)4 21.SOLUTIONa. 4x5 128x5COMMON ERRORWhen n is even and a 0,be sure to consider boththe positive and negativenth roots of a.Write original equation. 32Divide each side by 4.5—x 32Take fifth root of each side.x 2Simplify.The solution is x 2.b. (x 3)4 21Write original equation.4—x 3 21Take fourth root of each side.4—x 3 21Add 3 to each side.4—4—x 3 21 or x 3 21Write solutions separately.x 5.14Use a calculator.or x 0.86The solutions are x 5.14 and x 0.86.Real-Life ApplicationA hospital purchases an ultrasound machine for 50,000. The hospital expects theuseful life of the machine to be 10 years, at which time its value will have depreciatedto 8000. The hospital uses the declining balances method for depreciation, so theannual depreciation rate r (in decimal form) is given by the formulaS 1/nr 1 — .CIn the formula, n is the useful life of the item (in years), S is the salvage value (indollars), and C is the original cost (in dollars). What annual depreciation rate did thehospital use?()SOLUTIONThe useful life is 10 years, so n 10. The machine depreciates to 8000, so S 8000.The original cost is 50,000, so C 50,000. So, the annual depreciation rate is1/n()Sr 1 —C(8000 1 —50,000)1/101/10( )4 1 —25 0.167.The annual depreciation rate is about 0.167, or 16.7%.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind the real solution(s) of the equation. Round your answer to two decimalplaces when appropriate.13. 8x3 6414. —12 x5 51215. (x 5)4 1616. (x 2)3 1417. WHAT IF? In Example 5, what is the annual depreciation rate when the salvagevalue is 6000?196Chapter 4Int Math3 PE 0401.indd 196Rational Exponents and Radical Functions1/30/15 3:32 PM
4.1ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY Rewrite the expression a s/t in radical form. Then state the index of the radical.2. COMPLETE THE SENTENCE For an integer n greater than 1, if bn a, then b is a(n) of a.3. WRITING Explain how to use the sign of a to determine the number of real fourth roots of a and thenumber of real fifth roots of a.4. WHICH ONE DOESN’T BELONG? Which expression does not belong with the other three? Explainyour reasoning.n— m( a)(a1/n)mm — n( a)am/nMonitoring Progress and Modeling with MathematicsIn Exercises 5–10, find the indicated real nth root(s)of a. (See Example 1.)USING STRUCTURE In Exercises 21–24, match theequivalent expressions. Explain your reasoning.5. n 3, a 86. n 5, a 121.( 3 —5 )4A. 5 1/47. n 2, a 08. n 4, a 25622.( 4 —5 )3B. 54/39. n 5, a 3210. n 6, a 7291C. 51/423. ——4In Exercises 11–18, evaluate the expression withoutusing a calculator. (See Example 2.) 54—24. 5D. 53/411. 641/612. 81/313. 253/214. 813/4In Exercises 25–32, evaluate the expression using acalculator. Round your answer to two decimal placeswhen appropriate. (See Example 3.)15. ( 243)1/516. ( 64)4/325.17. 8 2/318. 16 7/427. 25 1/328. 851/629. 20,7364/530. 86 5/6ERROR ANALYSIS In Exercises 19 and 20, describe andcorrect the error in evaluating the expression.19.20. 31.272/3 (271/3)27—5—26. 1695 32,7683( 4 —187 )32.8( 5 — 8 )MATHEMATICAL CONNECTIONS In Exercises 33 and 34, 92find the radius of the figure with the given volume. 8133. V 216 ft32564/3 34. V 1332 cm3r3( 4 —256 )r 439 cm 64Section 4.1Int Math3 PE 0401.indd 197nth Roots and Rational Exponents1971/30/15 3:32 PM
In Exercises 35–44, find the real solution(s) of theequation. Round your answer to two decimal placeswhen appropriate. (See Example 4.)47. NUMBER SENSE Between which two consecutive35. x 3 12536. 5x3 108048. THOUGHT PROVOKING In 1619, Johannes Kepler37. (x 10)5 7038. (x 5)4 25639. x 5 4840. 7x 4 5641. x 6 36 10042. x 3 40 2543. —13 x 4 2744. —16 x 3 364—integers does 125 lie? Explain your reasoning.published his third law, which can be given byd 3 t 2, where d is the mean distance (in astronomicalunits) of a planet from the Sun and t is the time(in years) it takes the planet to orbit the Sun. It takesMars 1.88 years to orbit the Sun. Graph a possiblelocation of Mars. Justify your answer. (The diagramshows the Sun at the origin of the xy-plane and apossible location of Earth.)y45. MODELING WITH MATHEMATICS When the averageprice of an item increases from p1 to p2 over a periodof n years, the annual rate of inflation r (in decimalp2 1/nform) is given by r — 1. Find the rate ofp1inflation for each item in the table. (See Example 5.)(1, 0)x( )Price in1913Price in2013Potatoes (lb) 0.016 0.627Ham (lb) 0.251 2.693Eggs (dozen) 0.373 1.933ItemNot drawn to scale49. PROBLEM SOLVING A weir is a dam that is builtacross a river to regulate the flow of water. The flowrate Q (in cubic feet per second) can be calculatedusing the formula Q 3.367ℓh3/2, whereℓis thelength (in feet) of the bottom of the spillway and his the depth (in feet) of the water on the spillway.Determine the flow rate of a weir with a spillway thatis 20 feet long and has a water depth of 5 feet.46. HOW DO YOU SEE IT? The graph of y x n is shownin red. What can you conclude about the value of n?Determine the number of real nth roots of a. Explainyour reasoning.spillwayhyy a50. REPEATED REASONING The mass of the particles thata river can transport is proportional to the sixth powerof the speed of the river. A certain river normallyflows at a speed of 1 meter per second. What must itsspeed be in order to transport particles that are twiceas massive as usual? 10 times as massive? 100 timesas massive?xMaintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsSimplify the expression. Write your answer using only positive exponents. 51. 5 5442452. —7Write the number in standard form.53. (z2) 356. 4 10 257. 8.2 10 158. 6.93 106Chapter 4Rational Exponents and Radical FunctionsInt Math3 PE 0401.indd 198454.( 3x2 )—(Skills Review Handbook)55. 5 103198(Skills Review Handbook)1/30/15 3:32 PM
4.2Properties of Rational Exponentsand RadicalsEssential QuestionHow can you use properties of exponents tosimplify products and quotients of radicals?Reviewing Properties of ExponentsWork with a partner. Let a and b be real numbers. Use the properties of exponentsto complete each statement. Then match each completed statement with the propertyit illustrates.Statementa.a 2Property , a 0A. Product of Powersb. (ab)4 c.(a3)4 B. Power of a Power C. Power of a Productd. a3 a4 ae. — , b 0ba6f. —2 , a 0a()USING TOOLSSTRATEGICALLY 3 (8)E. Zero ExponentF. Quotient of Powersg. a0 , a 0To be proficient in math,you need to consider thetools available to help youcheck your answers. Forinstance, the followingcalculator screen shows3—3—3—that 4 2 and 8are equivalent.(3 (4))(3 (2))D. Negative Exponent3G. Power of a QuotientSimplifying Expressions withRational ExponentsWork with a partner. Show that you can apply the properties of integer exponentsto rational exponents by simplifying each expression. Use a calculator to check youranswers. a. 52/3 54/3b. 31/5 34/5c. (42/3)3d. (101/2)485/2e. —81/272/3f. —75/32Simplifying Products andQuotients of Radicals2Work with a partner. Use the properties of exponents to write each expression as asingle radical. Then evaluate each expression. Use a calculator to check your answers.— 3—— 3—b. 5 25a. 3 12 4d. —— 2 4—3—4—— 984—c. 27 3e. —4— 1024 625f. —3— 5Communicate Your Answer4. How can you use properties of exponents to simplify products and quotientsof radicals?5. Simplify each expression.— —a. 27 6Section 4.2Int Math3 PE 0402.indd 1993— 240b. —3— 15 c. (51/2 161/4)2Properties of Rational Exponents and Radicals1991/30/15 3:33 PM
4.2 LessonWhat You Will LearnUse properties of rational exponents to simplify expressions withrational exponents.Core VocabulVocabularylarrysimplest form of a radical,p. 201conjugate, p. 202like radicals, p. 202Previousproperties of integerexponentsrationalizing thedenominatorabsolute valueUse properties of radicals to simplify and write radical expressionsin simplest form.Properties of Rational ExponentsThe properties of integer exponents that you have previously learned can also beapplied to rational exponents.Core ConceptProperties of Rational ExponentsLet a and b be real numbers and let m and n be rational numbers, such that thequantities in each property are real numbers.Property NameCOMMON ERRORWhen you multiply powers,do not multiply theexponents. For example,32 35 310. Definition Example 5(1/2 3/2) 52 25Product of PowersPower of a Power(am)n amn(35/2)2 3(5/2 2) 35 243Power of a Product(ab)m ambm(16 9)1/2 161/2 91/2 4 3 12Negative Exponent1a m —,a 0am1136 1/2 — —361/2 6Zero Exponenta0 1, a 02130 1Quotient of Powers am n, a 0—n 4(5/2 1/2) 42 16—1/2Power of a Quotientamaa—b()manam nam51/2am —,b 0bm53/2 45/2427—64( ) 1/3 271/3 3 — —641/3 4Using Properties of ExponentsUse the properties of rational exponents to simplify each expression. b. (6 4c. (4 3 )a. 71/4 71/2 7(1/4 1/2) 73/41/21/3)25 1/55 (61/2)2 (41/3)2 6(1/2 2) 4(1/3 2) 61 42/3 6 42/3 1 [(4 3)5] 1/5 (125) 1/5 12[5 ( 1/5)] 12 1 —12551d. — — 5(1 1/3) 52/31/3551/3( ) [( ) ]421/3e. —61/32 426—1/3 2 (71/3)2 7(1/3 2) 72/3Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSimplify the expression. 1. 23/4 21/23.200Chapter 4Int Math3 PE 0402.indd 200( )201/253—1/2332. —1/4 4. (51/3 71/4)3Rational Exponents and Radical Functions1/30/15 3:33 PM
Simplifying Radical ExpressionsThe Power of a Product and Power of a Quotient properties can be expressed usingradical notation when m 1/n for some integer n greater than 1.Core ConceptProperties of RadicalsLet a and b be real numbers such that the indicated roots are real numbers, andlet n be an integer greater than 1.Property NameDefinition n— Examplen—3— 3—3—Product Propertyn—a b a b 4 2 8 2Quotient Property ———nabn— a,b 0— —n— b4— 1624 2 1622 81 3—4—4—Using Properties of RadicalsUse the properties of radicals to simplify each expression.3—a. 124— 18 12 18 216 63— 80b. — 4— 53—3— 805 16 2—44——Product Property of RadicalsQuotient Property of RadicalsAn expression involving a radical with index n is in simplest form when these threeconditions are met. No radicands have perfect nth powers as factors other than 1. No radicands contain fractions. No radicals appear in the denominator of a fraction.To meet the last two conditions, rationalize the denominator by multiplyingthe expression by an appropriate form of 1 that eliminates the radical from thedenominator.Writing Radicals in Simplest Form5—3— 7Write the expressions (a) 135 and (b) —in simplest form.5— 8SOLUTION3— 27 53—a. 135 27 53—3— 3 55— 75— 7 Factor out perfect cube.3—Product Property of RadicalsSimplify.5— 4b. — ——5—5—5— 8 8 4Make the radicand in the denominator a perfect fifth power.5— 28 —5— 32Product Property of Radicals5— 28 —2Section 4.2int math3 pe 0402.indd 201Simplify.Properties of Rational Exponents and Radicals2014/27/17 3:45 PM
For a denominator that is a sum or difference involving square roots, multiply both thenumerator and denominator by the conjugate of the denominator. The expressions——a b c d——a b c dandare conjugates of each other, where a, b, c, and d are rational numbers.Writing a Radical Expression in Simplest Form1Write —— in simplest form.5 3SOLUTION—5 3 5 3 5 311—— ——5 3——The conjugate of 5 3 is 5 3 .———1( 5 3 ) —— 252 ( 3 )Sum and Difference Pattern—5 3 —22Simplify.Radical expressions with the same index and radicand are like radicals. To add orsubtract like radicals, use the Distributive Property.Adding and Subtracting Like Radicals and RootsSimplify each expression.4—4—a. 10 7 10b. 2(81/5) 10(81/5)3—3—c. 54 2SOLUTION4—4—4—4—a. 10 7 10 (1 7) 10 8 10b. 2(81/5) 10(81/5) (2 10)(81/5) 12(81/5)3—3—3—c. 54 2 27 2 2 3 2 2 (3 1) 2 2 23—3—Monitoring Progress3—3—3—3—Help in English and Spanish at BigIdeasMath.comSimplify the expression.4—5. 273 4— 39. ——6 23— 2503 25—5—10. 7 12 12 34—3—7. 1046. ——8.5—3—3—11. 4(92/3) 8(92/3) 12. 5 40The properties of rational exponents and radicals can also be applied to expressionsinvolving variables. Because a variable can be positive, negative, or zero, sometimesabsolute value is needed when simplifying a variable expression.Rulen—nWhen n is odd x xWhen n is even xn x n—Example7—7— 57 5 and ( 5)7 54—4— 34 3 and ( 3)4 3Absolute value is not needed when all variables are assumed to be positive.202Chapter 4Int Math3 PE 0402.indd 202Rational Exponents and Radical Functions1/30/15 3:33 PM
Simplifying Variable ExpressionsSimplify each expression.a.STUDY TIPYou do not need totake the absolute valueof y because y isbeing squared. —3—6 64yb.4x4y—8SOLUTIONa. 64y6 43(y2)3 43 (y2)3 4y23— 3—b.3—44 x x 4x 4 x 4 —8 ———4—4—yy2y8( y 2)4——43—— Writing Variable Expressions in Simplest FormWrite each expression in simplest form. Assume all variables are positive.5—COMMON ERRORYou must multiply boththe numerator anddenominator of the3—fraction by y so thatthe value of the fractiondoes not change.14xy1/3c. —2x3/4z 6xb. —3— y 8a. 4a8b14c5SOLUTION5—5 ——a. 4a8b14c5 4a5a3b10b4c55— a5b10c5Factor out perfect fifth powers. 4a b5—3 4Product Property of Radicals5— ab2c 4a3b4x xSimplify.3— yb. — ——3—3—3— y 8 y 8 yMake denominator a perfect cube.—3x y —3— y 9Product Property of Radicals—3x y —y3Simplify.14xy1/3c. — 7x (1 3/4)y1/3z ( 6) 7x1/4 y1/3z62x3/4z 6Adding and Subtracting Variable ExpressionsPerform each indicated operation. Assume all variables are positive.—3——3—b. 12 2z5 z 54z2a. 5 y 6 ySOLUTION————a. 5 y 6 y (5 6) y 11 y3—3—3—3—3—3—b. 12 2z5 z 54z2 12z 2z2 3z 2z2 (12z 3z) 2z2 9z 2z2Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSimplify the expression. Assume all variables are positive.13.3—9 27qSection 4.2Int Math3 PE 0402.indd 203—14. 5x10y—56xy3/43x y15. —1/2 1/2——16. 9w5 w w3Properties of Rational Exponents and Radicals2031/30/15 3:33 PM
Exercises4.2Dynamic Solutions available at BigIdeasMath.comVocabulary and Core Conceptp Check1. WRITING How do you know when a radical expression is in simplest form?2. WHICH ONE DOESN’T BELONG? Which radical expression does not belong with the other three?Explain your reasoning.— 345——2 x4—5—3 9x 11Monitoring Progress and Modeling with MathematicsIn Exercises 3–12, use the properties of rationalexponents to simplify the expression. (See Example 1.)3. (92)1/34. (122)1/4667.1 1/48410—48. 39. (3 2/3 22/31/3) 1162/311. —2/34 1/3( )936 72 6 8——15.4—4——35— 486 5 )49 49—7 3/2 1/43/87/85/4 3—4—4—— 2 23— 6 7219. ——3 23—3— 3 1820. ———66In Exercises 21–28, write the expression in simplestform. (See Example 3.)4—21. 5675—22. 2883—4— 5 423. ——24. ——325. 38 6449——27.2043— 3 5In Exercises 37– 46, simplify the expression.(See Example 5.)3——Chapter 4Int Math3 PE 0402.indd 20428.43—6—6—37. 9 11 3 1138. 8 5 12 539. 3(111/4) 9(111/4)40. 13(83/4) 4(83/4)——5—5———42. 27 6 7 1503—3—43. 224 3 744. 7 2 12845. 5(241/3) 4(31/3)46. 51/4 6(4051/4)47. ERROR ANALYSIS Describe and correct the error insimplifying the expression. 74 1296253— 736. ——— 10 235. ————26.34. ——— 8 7 6 27—233. ——— 3 74 49 641. 5 12 19 3 325 23— 32 8 83— 218. ——17. ——32. ——914. 1616.1131. ——3 2In Exercises 13–20, use the properties of radicals tosimplify the expression. (See Example 2.)13. 22 5510. (51/212.30. ——1 36. —1/3( )129. ——775. —1/4In Exercises 29–36, write the expression in simplestform. (See Example 4.)3—3—3—3 12 5 12 (3 5) 243— 8 24 3— 8 8 3 3—— 8 2 3— 16 33——Rational Exponents and Radical Functions1/30/15 3:33 PM
48. MULTIPLE REPRESENTATIONS Which radicalexpressions are like radicals?A (52/9)3/2 3—C 625 3—53B E 5 3 5 3— 8( 5 )3—3—4—4—F 7 80 2 405 In Exercises 49–54, simplify the expression.(See Example 6.)49.3—4—50. 64r 3t 6 81y8 5m10n—552.—53.6 4 k1616zg6hh54.8——66. 11 2z 5 2z67. 3x7/2 5x7/23—68. 7 m7 3m7/34—4—69. 16w10 2w w6 p1/4)4— 16p3MATHEMATICAL CONNECTIONS In Exercises 71 and 72,—4find simplified expressions for the perimeter and area ofthe given figure.——7—70. (p1/2——51.—3365. 12 y 9 y—D 5145 875 3—In Exercises 65–70, perform the indicated operation.Assume all variables are positive. (See Example 8.)n18p7np—2 171.72.x355. ERROR ANALYSIS Describe and correct the error insimplifying the expression. 4x1/36——6 64h1264h12g ———66 g 6 6 —26 (h2)6 —6— g 673. MODELING WITH MATHEMATICS The optimumdiameter d (in millimeters) of the pinhole in a pinholecamera can be modeled by d 1.9[(5.5 10 4)ℓ]1/2,whereℓis the length (in millimeters) of the camerabox. Find the optimum pinhole diameter for a camerabox with a length of 10 centimeters.2h2 —gpinholefilm56. OPEN-ENDED Write two variable expressionsinvolving radicals, one that needs absolute value insimplifying and one that does not need absolute value.Justify your answers.In Exercises 57–64, write the expression in simplestform. Assume all variables are positive. (See Example 7.)—57. 81a7b12c9 3—58. 125r 4s9t7—59.5160m6n—73— — w w561. —— 25w1618w1/3v5/427w v63. —4/3 1/23x1/32x2/3 —60.4405x3y35x y— 14—tree74. MODELING WITH MATHEMATICS The surface area S(in square centimeters) of a mammal can be modeledby S km2/3, where m is the mass (in grams) of themammal and k is a constant. The table shows thevalues of k for different mammals.MammalValue of kRabbit Human9.7511.0Bat57.5 v662. —— v5a. Find the surface area of a bat whose mass is32 grams.7x 3/4 y5/2z 2/356xyb. Find the surface area of a rabbit whose mass is3.4 kilograms (3.4 103 grams).764. —— 1/2 1/4c. Which mammal has the greatest mass per squarecentimeter of surface area, the bat in part (a),the rabbit in part (b), or a human whose mass is59 kilograms?Section 4.2Int Math3 PE 0402.indd 205Properties of Rational Exponents and Radicals2051/30/15 3:33 PM
75. MAKING AN ARGUMENT Your friend —claims it —is notpossible to simplify the expression 7 11 9 4478. HOW DO YOU SEE IT? Without finding points, matchbecause it does not contain like radicals. Is your friendcorrect? Explain your reasoning.76. PROBLEM SOLVING The apparent magnitude of a tair0.77AquilaDeneb1.25Cygnusy16161212884 4 224x 4 224x79. REWRITING A FORMULA You have filled two roundballoons with water. One balloon contains twice asmuch water as the other balloon.a. Solve the formula for the volume of a sphere,V —43πr 3, for r.a. How many times fainter is Altair than Vega?b. How many times fainter is Deneb than Altair?b. Substitute the expression for r from part (a)into the formula for the surface area of a sphere,S 4πr2. Simplify to show that S (4π)1/3(3V )2/3.c. How many times fainter is Deneb than Vega?DenebB.yis a number that indicates how faint the star is in2.512m1relation to other stars. The expression —tells2.512m2how many times fainter a star with apparent magnitudem1 is than a star with apparent magnitude m2.Star3——the functions f(x) 64x 2 and g(x) 64x 6 withtheir graphs. Explain your reasoning.Vegac. Compare the surface areas of the two waterballoons using the formula in part (b).LyraCygnus80. THOUGHT PROVOKING Determine whether theexpressions (x2)1/6 and (x1/6)2 are equivalent for allvalues of x.AltairAquila81. DRAWING CONCLUSIONS Substitute differentcombinations of odd and even positive integers forn—m and n in the expression x m . When you cannotassume x is positive, explain when absolute value isneeded in simplifying the expression.77. CRITICAL THINKING Find a radical expression forthe perimeter of the triangle inscribed in the squareshown. Simplify the expression.282. REWRITING A FORMULA Rewrite the formula in4mExercise 74 so that one side is —. Use this formula toSjustify your answer in part (c).48Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsGraph the function. Label the vertex, axis of symmetry, and x-intercepts.83. g(x) 6(x 4)284. h(x) 2x(x 3)(Section 2.6)85. f (x) x 2 2x 5Write a rule for g. Describe the graph of g as a transformation of the graph of f.(Section 3.7)86. f(x) x4 3x2 2x, g(x) f(x)87. f(x) x3 x, g(x) f (x) 388. f(x) x3 4, g(x) f(x 2)89. f(x) x 4 2x 3 4x 2, g(x) f (2x)206Chapter 4Int Math3 PE 0402.indd 206Rational Exponents and Radical Functions1/30/15 3:33 PM
4.3Graphing Radical FunctionsEssential QuestionHow can you identify the domain and rangeof a radical function?Identifying Graphs of Radical FunctionsWork with a partner. Match each function with its graph. Explain your reasoning.Then identify the domain and range of each function.——a. f(x) x—3b. f(x) xA.B.4 646 4 4D.4 65d. f (x) x 66C.—4c. f(x) x4 666 4 4Identifying Graphs of Transformations—Work with a partner. Match each transformation of f (x) x with its graph.Explain your reasoning. Then identify the domain and range of each function.——a. g(x) x 2 b. g(x) x 2A.B.4 6—4 66 4C.LOOKING FORSTRUCTURETo be proficient in math,you need to look closelyto discern a pattern orstructure.6 4D.4 6—c. g(x) x 2 2 d. g(x) x 264 6 46 4Communicate Your Answer3. How can you identify the domain and range of a radical function?4. Use the results of Exploration 1 to describe how the domain and range of a radic
Find nth roots of numbers. Evaluate expressions with rational exponents. Solve equations using nth roots. nth Roots You can extend the concept of a square root to other types of roots. For example, 2 is a cube root of 8 because 23 8. In general, for an integer n greater than 1, if bn a, then b is an nth
Chapter 5: Rational Exponents and Radical Functions Date Section Topic HW Due Date 5.1 nth Roots, Radicals, Rational Exponents 5.2 Properties of Exponents and Radicals 5.3 Graphing Radical Functions 5.4 Solving Radical Equations 5.5 Function Operations 5.6 Inverse Relations and
Page 6 of 18 A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent. ( 2) 25 3 10 3 2 x x radical equations 3 x 10 NOT a radical equation Give your own: Radical equation Non radical equation To solve a radical equation: isolate the radical on one side of the equation and then raise both sides of the
304 Chapter 6 Rational Exponents and Radical Functions 6.3 Lesson WWhat You Will Learnhat You Will Learn Graph radical functions. Write radical functions. Graph parabolas and circles. Graphing Radical Functions A radical function contains a radical expression with the independent variable in the radicand. When the radical is a square root, the function is called a square root
Extend the properties of exponents to rational exponents. NVACS HSN.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3
Unit 4 Radical Expressions and Rational Exponents (chapter 7) Learning Targets: Properties of Exponents 1. I can use properties of exponents to simplify expressions. Simplifying Radical Expressions 2. I can simplify radical algebraic expressions. Multiplying and Dividing 3. I can multiply radical expressions. 4.
Radicals and Radical Functions . RD.2 Rational Exponents . In . sections P2. and . RT1, we reviewed properties of powers with natural and integral exponents. All of these properties hold for real exponents as well. In this section, we give meaning to expressions with rational e
5 Properties of Exponents 6 Radical Expressions and Expressions with Rational Exponents 7 Adding Expressions with Radicals and Rational Exponents 8 More Operations with Radicals and Rational Exponents
Rational Rational Rational Irrational Irrational Rational 13. 2 13 14. 0.42̅̅̅̅ 15. 0.39 16. 100 17. 16 18. 43 Rational Rational Rational Rational Rational Irrational 19. If the number 0.77 is displayed on a calculator that can only display ten digits, do we know whether it is rational or irrational?
