Radical Expressions, 10 Equations, And Functions
Ch10pgs677-6891/19/0612:03 PMPage 677Radical Expressions,Equations, andFunctionsReal-World ApplicationAn observation deck near the top of the Sears Towerin Chicago is 1353 ft high. How far can a tourist seeto the horizon from this deck?This problem appearsas Exercise 45 inSection 10.6.10.110.210.310.410.510.610.7Radical Expressions and FunctionsRational Numbers as ExponentsSimplifying Radical ExpressionsAddition, Subtraction, andMore MultiplicationMore on Division of RadicalExpressionsSolving Radical EquationsApplications Involving Powersand RootsThe Complex NumbersISBN:0-536-47742-610.810Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs677-6891/19/0612:03 PMPage 67810.1ObjectivesFind principal square rootsand their opposites,approximate square roots,find outputs of squareroot functions, graphsquare-root functions, andfind the domains ofsquare-root functions.Simplify radicalexpressions with perfectsquare radicands.Find cube roots, simplifyingcertain expressions, andfind outputs of cube-rootfunctions.Simplify expressionsinvolving odd andeven roots.RADICAL EXPRESSIONSAND FUNCTIONSIn this section, we consider roots, such as square roots and cube roots. Wedefine the symbolism and consider methods of manipulating symbols to getequivalent expressions.Square Roots and Square-Root FunctionsWhen we raise a number to the second power, we say that we have squaredthe number. Sometimes we may need to find the number that was squared.We call this process finding a square root of a number.SQUARE ROOTThe number c is a square root of a if c 2 a.For example:5 is a square root of 25 because 52 5 5 25; 5 is a square root of 25 because 5 2 5 5 25.The number 4 does not have a real-number square root because thereis no real number c such that c 2 4.Find the square roots.1. 9PROPERTIES OF SQUARE ROOTSEvery positive real number has two real-number square roots.The number 0 has just one square root, 0 itself.2. 36Negative numbers do not have real-number square roots.*EXAMPLE 1Find the two square roots of 64.The square roots of 64 are 8 and 8 because 82 64 and 8 2 64.3. 121Do Exercises 1–3.PRINCIPAL SQUARE ROOTSimplify.4. 16. 81100The principal square root of a nonnegative number is its nonnegativesquare root. The symbol a represents the principal square root of a.To name the negative square root of a, we can write a.5. 36EXAMPLESSimplify.7. 0.0064Remember: indicates the principal(nonnegative) square root.3. 25 5Answers on page A-43678CHAPTER 10: Radical Expressions,Equations, and Functions*In Section 10.8, we will consider a number system in which negative numbers do havesquare roots.Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.ISBN:0-536-47742-62. 25 5
Ch10pgs677-6891/19/064.12:03 PMPage 679 Find the following.819 6488. a) 169. a) 495. 0.0049 0.076. 0.000001 0.001b) 16b) 49c) 16c) 497. 0 08. 25 Does not exist as a real number. Negative numbers do not havereal-number square roots.Do Exercises 4–13. (Exercises 4–7 are on the preceding page.)10. a) 144We found exact square roots in Examples 1– 8. We often need to userational numbers to approximate square roots that are irrational. Such expressions can be found using a calculator with a square-root key.EXAMPLESUsing a calculator witha 10-digit readout9. 113.31662479010. 48711. 7297.812. 463557Rounded to threedecimal places22.068 85.42716196 85.427.911722972812. 0.810.912TABLE OF COMMONSQUARE ROOTS 1 1 4 2 9 3 16 4 25 5 36 6 49 7 64 8 81 9 100 10 121 11 144 12 169 13RADICAL; RADICAL EXPRESSION;RADICANDis called a radical.An expression written with a radical is called a radical expression.The expression written under the radical is called the radicand.These are radical expressions: 5 , a , 5x , y 2 7 .The radicands in these expressions are 5, a, 5x, and y 2 7, respectively.EXAMPLE 1313. 1.44It would be helpful to memorize thefollowing table of exact square roots.Do Exercises 14–19.The symbol 2564c) 1443.31722.06807649 b) 144Use a calculator to approximate each of the following.Number11. 196 14 225 15 256 16 289 17 324 18 361 19 400 20 441 21 484 22 529 23 576 24 625 25Identify the radicand in x 2 9 .The radicand in x 2 9 is x 2 9.ISBN:0-536-47742-6Do Exercises 20 and 21 on the following page.Use a calculator to approximate thesquare root to three decimal places.14. 1715. 4016. 113817. 867.618. 2235 19. 2103.467.82Answers on page A-4367910.1 Radical Expressions and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs677-6891/19/0612:03 PMIdentify the radicand.Page 680Since each nonnegative real number x has exactly one principal squareroot, the symbol x represents exactly one real number and thus can be usedto define a square-root function:20. 28 xf x x .The domain of this function is the set of nonnegative real numbers. In interval notation, the domain is 0, . This function will be discussed further inExample 16.EXAMPLE 14For the given function, find the indicated function values:f x 3x 2 ;21.f 1 , f 5 , and f 0 .We have yy 3f 1 3 1 2Substituting 1 for x 3 2 1 1;f 5 3 5 2 13Substituting 5 for x3.606;f 0 3 0 2 2 .Simplifying and taking the square rootSimplifying and approximatingSubstituting 0 for xNegative radicand. No real-number functionvalue exists; 0 is not in the domain of f.Do Exercises 22 and 23.EXAMPLE 15For the given function, find theindicated function values.22. g x 6x 4; g 0 , g 3 ,and g 5 Find the domain of g x x 2 .The expression x 2 is a real number only when x 2 is nonnegative. Thus the domain of g x x 2 is the set of all x-values for whichx 2 0. We solve as follows:x 2 0x 2.Adding 2The domain of g x x 2 2, .EXAMPLE 16Graph: (a) f x x ; (b) g x x 2 .We first find outputs as we did in Example 14. We can either select inputsthat have exact outputs or use a calculator to make approximations. Onceordered pairs have been calculated, a smooth curve can be drawn.a)23. f x x;f 3 f 4 , f 7 , andyxf x xx, f x013479011.722.63 0, 0 1, 1 3, 1.7 4, 2 7, 2.6 9, 3 54f (x) œx(9, 3)(3, 1.7)2(7, 2.6)(1, 1)(4, 2)1(0, 0)3 3 2 1 11 2 3 4 5 6 7 8 9x 2Answers on page A-43680CHAPTER 10: Radical Expressions,Equations, and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.ISBN:0-536-47742-6We can see from the table and the graph that the domain is 0, .The range is also the set of nonnegative real numbers 0, .
Ch10pgs677-689b)1/19/0612:03 PMPage 681g x x 2x 2 103510011.42.22.63.5Find the domain of the function.yx, g x 2, 0 1, 1 0, 1.4 3, 2.2 5, 2.6 10, 3.5 543g (x) x 2(3, 2.2)2( 1, 1)24. f x x 5(10, 3.5)(5, 2.6)1 (0, 1.4) 2 1 11 2 3 4 5 6 7 8 9 10x( 2, 0) 225. g x 2x 3We can see from the table, the graph, and Example 15 that the domain is 2, . The range is the set of nonnegative real numbers 0, .Do Exercises 24–27.Graph.Finding a226. g x xIn the expression a 2 , the radicand is a perfect square. It is tempting to thinkthat a 2 a, but we see below that this is not the case.y5Suppose a 5. Then we have 52 , which is 25 , or 5.4Suppose a 5. Then we have 5 2 , which is 25 , or 5.2Suppose a 0. Then we have 0 2 ,which is 0 , or 0.The symbol a 2 never represents a negative number. It represents theprincipal square root of a 2. Note the following.31 3 2 1 11 2 3 4 5 6 7 8 9x 2 3 4 5SIMPLIFYING a2a 0 a 2 aIf a is positive or 0, the principal square root of a 2 is a.a 0 a 2 aIf a is negative, the principal square root of a 2 is the opposite of a.27. f x 2 x 3In all cases, the radical expression represents the absolute value of a.PRINCIPAL SQUARE ROOT OF a2For any real number a, a . The principal (nonnegative) squareroot of a 2 is the absolute value of a. a 2y54321 3 2 1 1The absolute value is used to ensure that the principal square root isnonnegative, which is as it is defined.1 2 3 4 5 6 7 8 9x 2 3 4ISBN:0-536-47742-6 5Answers on page A-4368110.1 Radical Expressions and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs677-6891/19/0612:03 PMFind the following. Assume thatletters can represent any realnumber.28. y 2Page 682EXAMPLESFind the following. Assume that letters can represent anyreal number.17. 16 2 16 , or 1618. 3b 2 3b 3 b 3 b 29. 24 2 3b can be simplified to 3 b because the absolute value of any product isthe product of the absolute values. That is, a b a b .19. x 1 2 x 1 30. 5y 220. x 2 8x 16 x 4 2 x 4 31. 16y 2Caution! x 4 is not thesame as x 4.Do Exercises 28–35.Cube Roots32. x 7 2CUBE ROOT333. 4 x The number c is the cube root of a, written a, if the third power of c3is a—that is, if c 3 a, then a c.2 234. 49 y 5 2For example:2 is the cube root of 8 because 23 2 2 2 8; 4 is the cube root of 64 because 4 3 4 4 4 64.35. x 2 6x 9We talk about the cube root of a number rather than a cube root because ofthe following.Every real number has exactly one cube root in the system of real3numbers. The symbol a represents the cube root of a.Find the following.336. 64EXAMPLES337. 27y 33322. 27 32166 125524. 0.001 0.1 3 26. 8 2 Answers on page A-433328. 8y 3 2y 3 2y33When we are determining a cube root, no absolute-value signs areneeded because a real number has just one cube root. The real-numbercube root of a positive number is positive. The real-number cube root of a3negative number is negative. The cube root of 0 is 0. That is, a 3 awhether a 0, a 0, or a 0.Do Exercises 36–39.682CHAPTER 10: Radical Expressions,Equations, and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.ISBN:0-536-47742-639.343 643325. x 3 x27. 0 03321. 8 2 because 23 8.23.38. 8 x 2 3Find the following.
Ch10pgs677-6891/19/0612:03 PMPage 6833Since the symbol x represents exactly one real number, it can be used3to define a cube-root function: f x x .40. For the given function, find theindicated function values:3For the given function, find the indicated function values:EXAMPLE 293f x x ;g x x 4 ; g 23 ,g 4 , g 1 , and g 11 .f 125 , f 0 , f 8 , and f 10 .We have3f 125 125 5;3f 0 0 0;Find the following.3f 8 8 2;3f 10 10541. 243 2.1544.For calculator instructions for findinghigher roots, see the Calculator Corneron p. 685.Do Exercise 40.53The graph of f x x is shown below for reference. Note that the domain and the range each consists of the entire set of real numbers, , .42. 243y3f (x) œx43(0, 0) 8 7 6 5 4 3 2 1( 8, 2)( 1, 1)5(8, 2)43. x 52 (1, 1)11 2 3 4 5 6 7 8x 2 3 4744. y 7Odd and Even kth RootskIn the expression a, we call k the index and assume k 2.5ODD ROOTSThe 5th root of a number a is the number c for which c 5 a. There are alsok7th roots, 9th roots, and so on. Whenever the number k in is an oddnumber, we say that we are taking an odd root.3Every number has just one real-number odd root. For example, 8 2,33 8 2, and 0 0. If the number is positive, then the root is positive.If the number is negative, then the root is negative. If the number is 0, then theroot is 0. Absolute-value signs are not needed when we are finding odd roots.45. 0546. 32x 5If k is an odd natural number, then for any real number a,ISBN:0-536-47742-6k k a. a747. 3x 2 7Answers on page A-4368310.1 Radical Expressions and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs677-6891/19/0612:03 PMFind the following. Assume thatletters can represent any realnumber.448. 81Page 684EXAMPLESFind the following.5530. 32 231. 32 25533. 32 2 232. 32 2735. 128 27737. 0 034. x 7 x736. 128 25938. a 5 a39. x 1 9 x 1449. 81Do Exercises 41–47 on the preceding page.EVEN ROOTS450. 81kWhen the index k in is an even number, we say that we are taking an evenroot. When the index is 2, we do not write it. Every positive real number hastwo real-number kth roots when k is even. One of those roots is positive andone is negative. Negative real numbers do not have real-number kth rootswhen k is even. When we are finding even kth roots, absolute-value signs aresometimes necessary, as they are with square roots. For example, 64 8,451. 06 64 2,6 64 2,66 64x 6 2x 6 2x 2 x .6Note that in 64x 6 , we need absolute-value signs because a variable isinvolved.EXAMPLESFind the following. Assume that letters can represent anyreal number.452. 16 x 2 4440. 16 2441. 16 2442. 16 Does not exist as a real number.4443. 81x 4 3x 4 3 x 6653. x 644. y 7 6 y 7 45. 81y 2 9y 2 9 y The following is a summary of how absolute value is used when we aretaking even or odd roots.854. x 3 8kSIMPLIFYING akFor any real number a:755. x 3 7k k a when k is an even natural number. We use absolutea) avalue when k is even unless a is nonnegative.k kb) a a when k is an odd natural number greater than 1. We donot use absolute value when k is odd.Do Exercises 48–56.556. 243x 5ISBN:0-536-47742-6Answers on page A-43684CHAPTER 10: Radical Expressions,Equations, and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs677-6891/19/0612:04 PMPage 685CALCULATOR CORNERApproximating Roots We can use a graphing calculator to approximate square roots, cube roots, and higherroots of real numbers. To approximate 21, for example, we press F 2 1 ) [ . ( is thesecond operation associated with the V key.) To approximate 6.95 , we press : F 6 . 9 5) [ . Although it is not necessary to include the right parenthesis in either of these entries, we do so here inorder to close the set of parentheses that are opened when the calculator displays “ ”. We see that 21 4.583 and 6.95 2.636.3We can also find higher roots on a graphing calculator. To find 71, we will use the cube-root operation fromthe MATH menu. We press L 4 to select this operation. Then we press : 7 1 ) [ to enter theradicand and display the result. As with square roots, we choose to close the parentheses although it is not necessaryfor this calculation. To find fourth, fifth, or higher roots, we use the xth-root operation from the MATH menu. To find6, we first press 6 to indicate that we are finding a sixth root. Then we press L 5 to select the xth-root 178.4operation. Finally, we press 1 7 8 . 4 [ to enter the radicand and display the result. Note that since thisoperation does not supply a left parenthesis, we do not enter a right parenthesis at the end. We see that36 4.141 and 178.42.373. 713 (21) (6.95)4.582575695 2.636285265( 71)6 178.4 4.1408177492.372643426Exercises: Use a graphing calculator to approximate each of the following to three decimal places.1. 432. 10,46735. 416.7336. 8009. 3 510. 3 5411. 3 7712. 3 77. 16.44. 11 178. 1389.7ISBN:0-536-47742-63. 940668510.1 Radical Expressions and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs677-6891/19/0610.112:04 PMPage 686EXERCISE SETMathXLFor Extra HelpMyMathLabInterActMathMath Tutor Digital VideoCenterTutor CD 5Videotape 10Student’sSolutionsManualFind the square roots.1. 162. 225Simplify. 7. 3. 144 49368. 11. 0.0036361912. 0.044. 910. 44113. 22514. 6419. 9 y 2 16 2857416. 1839.217.20. 3 p 2 1021. x 4y 5Identify the radicand.6. 819. 196Use a calculator to approximate to three decimal places.15. 3475. 40018. xy 1 839.419.7 22. a 2b 2a2 bbFor the given function, find the indicated function values.23. f x 5x 10 ;f 6 , f 2 , f 1 , and f 1 24. t x 2x 1 ; t 4 , t 0 , t 1 , and t 12 25. g x x 2 25 ; g 6 , g 3 , g 6 , and g 13 26. F x x 2 1 ; F 0 , F 1 , and F 10 27. Find the domain of the function f in Exercise 23.28. Find the domain of the function t in Exercise 24.29. Speed of a Skidding Car. How do police determinehow fast a car had been traveling after an accident hasoccurred? The function30. Parking-Lot Arrival Spaces. The attendants at aparking lot park cars in temporary spaces beforethe cars are taken to permanent parking stalls. Thenumber N of such spaces needed is approximatedby the functionS x 2 5xN a 2.5 a ,where a is the average number of arrivals in peak hours.What is the number of spaces needed when the averagenumber of arrivals is 66? 100?686CHAPTER 10: Radical Expressions,Equations, and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.ISBN:0-536-47742-6Copyright 2007 Pearson Education, Inc.can be used to approximate the speed S, in miles perhour, of a car that has left a skid mark of length x, infeet. What was the speed of a car that left skid marksof length 30 ft? 150 ft?
Ch10pgs677-6891/19/0612:04 PMPage 687Graph.31. f x 2 x32. g x 3 xyy54321 5 4 3 2 1 1 2 3 4 51 2 3 4 5x 5 4 3 2 1 1 2 3 4 5x 5 4 3 2 1 1 2 3 4 51 2 3 4 5x 5 4 3 2 1 1 2 3 4 51 2 3 4 5x 5 4 3 2 1 1 2 3 4 5x 5 4 3 2 1 1 2 3 4 5 5 4 3 2 1 1 2 3 4 51 2 3 4 5x 5 4 3 2 1 1 2 3 4 51 2 3 4 5x42. f x 3x 6y543211 2 3 4 5x 5 4 3 2 1 1 2 3 4 51 2 3 4 5Find the following. Assume that letters can represent any real number.ISBN:0-536-47742-6x5432154321x1 2 3 4 5yy1 2 3 4 5 5 4 3 2 1 1 2 3 4 538. g x x 341. g x 3x 9543211 2 3 4 5x54321y543211 2 3 4 5y40. g x 8 4xy5432137. f x x 25432139. f x 12 3x 5 4 3 2 1 1 2 3 4 51 2 3 4 5y54321y5432136. g x xy34. f x 2 x 1y5432135. f x x 5 4 3 2 1 1 2 3 4 533. F x 3 x43. 16x 244. 25t 245. 12c 246. 9d 247. p 3 248. 2 x 249. x 2 4x 450. 9t 2 30t 25687Exercise Set 10.1Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.x
Ch10pgs677-6891/19/0612:04 PMPage 688Simplify.352. 64356. 100051. 2755. 216353. 64x 3354. 125y 33357. 0.343 x 1 3358. 0.000008 y 2 33For the given function, find the indicated function values.359. f x x 1 ;33f 7 , f 26 , f 9 , and f 65 61. f x 3x 1 ;60. g x 2x 1 ; g 62 , g 0 , g 13 , and g 63 f 0 , f 7 , f 21 , and f 333 362. g t t 3 ; g 30 , g 5 , g 1 , and g 67 Find the following. Assume that letters can represent any real number.464. 256 567.670. y 866. 3269. x 658 32243565. 168. 5 132471. 5a 4CHAPTER 10: Radical Expressions,Equations, and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.ISBN:0-536-47742-6688Copyright 2007 Pearson Education, Inc.463. 625
Ch10pgs677-6891/19/0612:04 PMPage 68910472. 7b 4414199975. a b 414777. y 776. 2a b 19993578. 6 381.1274. 10 1273. 6 10979. x 2 5DW Does the nth root of x 2 always exist? Why or80. 2xy 982.why not?DW Explain how to formulate a radical expression thatcan be used to define a function f with a domain of x x 5 .SKILL MAINTENANCESolve. [5.8b]83. x 2 x 2 084. x 2 x 085. 4x 2 49 086. 2x 2 26x 72 087. 3x 2 x 1088. 4x 2 20x 25 089. 4x 3 20x 2 25x 090. x 3 x 2 0Simplify.91. a 3b 2c 5 3 [4.2a]92. 5a 7b 8 2a 3b [4.1d]SYNTHESIS93. Find the domain off x 94. x 3 2 xUse a graphing calculator to check your answersto Exercises 35, 39, and 41.95. Use only the graph of f x x , shown below, toapproximate 3 , 5 , and 10 . Answers may vary.y396. Use only the graph of f x x , shown below,333to approximate 4 , 6 , and 5 . Answersmay vary.y53f (x) œx4f (x) œx4332211 3 2 1 11 2 3 4 5 6 7 8 9x 8 7 6 5 4 3 2 1 21 2 3 4 5 6 7 8x 2 3 4ISBN:0-536-47742-697.Use the TABLE, TRACE, and GRAPH features of a graphing calculator to find the domain and the range of each of thefollowing functions.3a) f x xc) q x 2 x 34e) t x x 33b) g x 4x 54d) h x x689Exercise Set 10.1Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs690-6961/19/0612:07 PM10.2ObjectivesWrite expressions with orwithout rational exponents,and simplify, if possible.Write expressions withoutnegative exponents, andsimplify, if possible.Use the laws of exponentswith rational exponents.Use rational exponentsto simplify radicalexpressions.Rewrite without rational exponents,and simplify, if possible.1. y 1/4Page 6902. 3a 1/2RATIONAL NUMBERS AS EXPONENTSIn this section, we give meaning to expressions such as a 1/3, 7 1/2, and 3x , which have rational numbers as exponents. We will see that usingsuch notation can help simplify certain radical expressions.0.84Rational ExponentsExpressions like a 1/2, 5 1/4, and 2y 4/5 have not yet been defined. We will define such expressions so that the general properties of exponents hold.Consider a 1/2 a 1/2. If we want to multiply by adding exponents, it mustfollow that a 1/2 a 1/2 a 1/2 1/2, or a 1. Thus we should define a 1/2 to be asquare root of a. Similarly, a 1/3 a 1/3 a 1/3 a 1/3 1/3 1/3, or a 1, so a 1/33should be defined to mean a.a1/nFor any nonnegative real number a and any natural number index n n 1 ,na 1/n means a (the nonnegative nth root of a).1/41/34. 125 3. 16Whenever we use rational exponents, we assume that the bases arenonnegative.Rewrite without rational exponents, and simplify, if possible.EXAMPLES1. 2731/3 27 31/52. abc 5 abc3. x 1/2 x3 2An index of 2 is not written.1/55. a b c Do Exercises 1–5.EXAMPLESRewrite with rational exponents.36. 19ab3Rewrite with rational exponents.54. 7xy 7xy 1/5We need parentheses aroundthe radicand here.7. 19 ab35. 8 xy 8 xy 1/36. 7x 3y 9 x 3y91/7Do Exercises 6–9.8. 5x 2y1649. 7 2abHow should we define a 2/3 ? If the general properties of exponents are33to hold, we have a 2/3 a 1/3 2, or a 2 1/3, or a 2, or a 2. We define thisaccordingly.Answers on page A-43For any natural numbers m and n n 1 and any nonnegative realnumber a,na m/n means a m, or690CHAPTER 10: Radical Expressions,Equations, and Functions n a m.Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.ISBN:0-536-47742-6am/n
Ch10pgs690-6961/19/0612:07 PMPage 691Rewrite without rational exponents, and simplify, if possible.EXAMPLES327. 27 2/3 2728. 43/2 43 27 2 4 332 32 23 9 8Rewrite without rational exponents,and simplify, if possible.10. x 3/511. 82/3Do Exercises 10–12.Rewrite with rational exponents.EXAMPLESThe index becomes the denominator of the rational exponent.12. 45/210. 7xy 5 7xy 5/4349. 9 4 9 4/3Do Exercises 13 and 14.Rewrite with rational exponents.Negative Rational Exponents13. 7abc 43514. 67Negative rational exponents have a meaning similar to that of negative integer exponents.a m/nFor any rational number m n and any positive real number a,1,a m/na m/n meansRewrite with positive exponents,and simplify, if possible.that is, a m/n and a m/n are reciprocals.13. 64 2/3 1 5xy 4/518. 7p 3/4q 6/51111 3 2 2642/3 41664 14. 4x 2/3y 1/5 4 1x 3r7s17. 81 3/4111 9 1/2 9312. 5xy 4/5 15.16. 3xy 7/8Rewrite with positive exponents, and simplify, if possible.EXAMPLES11. 9 1/2 15. 16 1/4 5/2 7s3r1/5 2/3 y5/2Since4y 1/5x 2/3 ab n ban19.ISBN:0-536-47742-6Do Exercises 15–19. 11m7n 2/3Answers on page A-4369110.2 Rational Numbers as ExponentsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs690-6961/19/0612:07 PMPage 692CALCULATOR CORNERRational Exponents We can use a graphing calculator to approximaterational roots of real numbers. To approximate 72/3, we press 7 U (2 d 3 ) [ . Note that the parentheses around the exponentare necessary. If they are not used, the calculator will read the expression as72 3. To approximate 14 1.9, we press 1 4 U : 1 . 9 [ .Parentheses are not required when a rational exponent is expressed in a singledecimal number. The display indicates that 72/3 3.659 and 14 1.9 0.007.7 (2/3)14 1.93.65930571.006642885Exercises: Approximate each of the following.1. 53/44. 730.562. 84/75. 34 2.783. 29 3/86. 320.2Use the laws of exponents tosimplify.20. 71/3 73/5Laws of ExponentsThe same laws hold for rational-number exponents as for integer exponents.We list them for review.For any real number a and any rational exponents m and n:1. a m a n a m n21.57/655/62.am a m nan3. a m n a m n4. ab m a mb m5. abn anbn22. 93/5 2/3EXAMPLESIn multiplying, we can add exponents if thebases are the same.In dividing, we can subtract exponents if thebases are the same.To raise a power to a power, we can multiplythe exponents.To raise a product to a power, we can raiseeach factor to the power.To raise a quotient to a power, we can raiseboth the numerator and the denominator tothe power.Use the laws of exponents to simplify.16. 31/5 33/5 31/5 3/5 34/5Adding exponents1/423. p 2/3q1/4 1/2Subtracting exponents18. 7.22/3 3/4 7.22/3 3/4 7.26/12 7.21/2Multiplying exponents19. a 1/3b 2/5 1/2 a 1/3 1/2 b 2/5 1/2Raising a product to a powerand multiplying exponents a 1/6b 1/5 Answers on page A-43b 1/5a 1/6Do Exercises 20–23.692CHAPTER 10: Radical Expressions,Equations, and FunctionsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.ISBN:0-536-47742-617 71/4 1/2 71/4 2/4 7 1/4 1/471/2717.
Ch10pgs690-6961/19/0612:07 PMPage 693Use rational exponents to simplify.Simplifying Radical Expressions424. a 2Rational exponents can be used to simplify some radical expressions. Theprocedure is as follows.SIMPLIFYING RADICAL EXPRESSIONS1. Convert radical expressions to exponential expressions.2. Use arithmetic and the laws of exponents to simplify.3. Convert back to radical notation when appropriate.425. x 4Important: This procedure works only when all expressions underradicals are nonnegative since rational exponents are not definedotherwise. With this assumption, no absolute-value signs will beneeded.626. 8Use rational exponents to simplify.EXAMPLES620. x 3 x 3/6 xConverting to an exponential expression1/2Simplifying the exponent xConverting back to radical notation621. 4 41/6Converting to exponential notationUse rational exponents to simplify. 22 1/6Renaming 4 as 2227. x 3y 6 22/6Using a m n a mn; multiplying exponents 21/3Simplifying the exponent3 212Converting back to radical notationDo Exercises 24–26.6EXAMPLE 22Use rational exponents to simplify:8 a 2b 4 a 2b 4 1/8 a2/8 a1/4 a1/48 a 2b 4.28. a 12b 3Converting to exponential notation b4/8Using ab n a nb n b1/2Simplifying the exponents b2/4Rewriting 2 with a denominator of 4 ab 2 1/41Using a nb n ab n4 ab 2529. a 5b 10Converting back to radical notationDo Exercises 27–29.We can use properties of rational exponents to write a single radical expression for a product or a quotient.EXAMPLE 23Use rational exponents to write a single radical expression3for 5 2.3ISBN:0-536-47742-6 5 2 51/3 21/22/6 53/6 2 52 23 1/6Rewriting so that exponents have acommon denominatorUsing a nb n ab nConverting back to radical notation6Multiplying under the radical 2004 7 3.Converting to exponential notation6 52 2330. Use rational exponents to writea single radical expression:Answers on page A-4369310.2 Rational Numbers as ExponentsIntroductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright 2007 by Pearson Education, Inc.
Ch10pgs690-6961/19/0612:07 PMWrite a single radical expression.31. xPage 694Do Exercise 30 on the preceding page.2/3 1/2 5/6yzWrite a single radical expression for a 1/2b 1/2c 5/6.EXAMPLE 24a 1/2b 1/2c 5/6 a 3/6b 3/6c 5/6 a 3b 3c 5 1/66 a 3b 3c 5a1/2b3/832. 1/4 1/8a bx 5/6 y 3/8 x 5/6 4/9 y 3/8 1/4x 4/9 y 1/4Converting to radical notationx 5/6 y 3/8.x 4/9 y 1/4Subtracting exponents x 15/18 8/18 y 3/8 2/8Finding common denominators so thatexponents can be subtracted x 7/18 y 1/8Carrying out the subtraction ofexponents x
Find the square roots. 1. 9 2. 36 3. 121 Simplify. 4. 1 5. 36 6. 7. 0.0064 81 100 Answers on page A-43 Objectives Find principal square roots and their opposites, approximate square roots, find outputs of square-root functions, graph square-root functions, and find the domains of square-root functions. Simplify radical expressions with perfect .
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
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.
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
Learning Goal Check Students will be able to simplify and perform operations with radical expressions. Scale Rubric # of students 4 I can simplify complex radical expressions and teach levels 1-3 to a peer! 3 I use perform operations with radical expressions. 2 I can simplify radical expressions with numbers and variables. 1 I can simpli
Radical Expressions 8th Day 8 - Simplifying Radical Expressions Priority Standards Test 9th Quiz over Days 4 – 7 Boot Camp 12th Day 9 - Multiplying & Simplifying Radical Expressions 13th Teacher Work Day 14th Day 10 – Practice Multiplying & Simplifying Radical Expressions 15th Day 11 – Adding & Subtracting Radicals Priority Standards Test .
Expressions p. 157 Embedded Assessment 2: Expressions and Equations p. 211 Why are tables, graphs, and equations useful for representing relationships? How can you use equations to solve real-world problems? Unit Overview In this unit you will use variables to write expressions and equations. You will solve and graph equations and inequalities.
R.6 Radical Expressions and Equations Our goal in this section is to merely provide a brief review of radicals and radical equations. For a more expanded discussion, refer to a full treatment of radicals in chapter 8. We begin by simply reviewing the definition and simplifying of radicals. Definition: If a!0 and n is positive, then n a
Simplifying Radicals. Day 11 Simplifying Radical Expressions Notes.notebook 9 December 02, 2019 Simplifying Radicals Graphic Organizer. Day 11 Simplifying Radical Expressions Notes.notebook 10 December 02, 2019 Practice - I do. Day 11 Simplifying Radical Exp
