26 ChapterP r rn Prerequisites

Rational Exponentsand Radicals

Raising a number to a power is reversedby finding the root of a number. we indicate roots by using rational exponentsor radicals. In this section we will review defini- tions and rules concerning rational exponentsand radicals.

Roots

Since2a : 16 and (-D4 : 16,both 2 and -2 are fourth roots of 16.The nth root of a number is defined in terms of the nth Dower.

Definition:nth Roots lf n is a positive integer and an : b, then a is called an nth root of b. If a2 : b, thena is a squareroot ofD. lf a3 : b,thena is the cube root ofb.

we also describeroots as evenor odd"depending on whetherthe positive integeris even or odd. For example,iffl is even (or odd) and a is an nth root ofb, then a is called an even (or odd) root of b. Every positive real number has two real even roots, a positive root and a negativeroot. For example,both 5 and -5 are square roots of 25 because52 :25 and (-5)2 : 25. Moreover,every real numberhas exactly onereal odd root. For example,because 23 : 8 and 3 is odd,2 is the only real cuberoot of 8. Because(-2)3 : -8 and 3 is odd, -2 is the only real cube root of -8. Finding an nth root is the reverseof finding an nth power, so we use the notation at/n for the nth root of a. For example, since the positive squareroot of 25 is 5, we write25t/2 : 5.

Definition:Exponent 1 /n If z is a positive even integer and a is positive, then at/n denotes the positive real zth root ofa and is called the principal zth root ofa" If n is a positive odd integer and a is any real number, then arln denotesthe real nth root ofa. If n is a positive integer,then 0lln : 0.

fua.*V/e I rvaluatingexpressions involving exponent 1/n Evaluate each expression.

a. 4tl2 b. gr/3 c. (-g;t/: d. (+yrlz

Solution a. The expression4ll2 representsthe positivereal squareroot of 4. So 4tl2 : 2. b. 8t/3 : 2 c. (_g)t/3: _2 P.3r ilffi RationalExoonents and Radicals 27

d. Since the definition of nth root does not include an even root of a negative num- ber, (_471/zhas not yet beendefined. Even roots of negativenumbers do exist in the complex number system,which we define in Section P.4.So an even root of a negative number is not a real number.

TrVThs.Evaluate. a. 91/2 b. l6tl4

Rational Exponents We havedefined otl" asthe nthroot of a. We extendthis definitionto e'ln,which is defined as the mth power of the nth root of a. A rational exponent indicates both a root and a power.

Definition: If m andr arepositive integers, then RationalExponents e^ln : @rlr)^ providedthat all' is a realnumber.

Note that a'tn is not real when a is negative and n is even. According to the defini- tion of rationalexponents, expressions such as (_251-zlz, (*+31t1+,and(- l)21'ate not defined becauseeach ofthem involves an even root ofa nesative number. Note thatsomeauthorsdefinea'l' onlyfor mfninlowestterms.In-thatcasethefourth power of the square root of 3 could not be written as 3a/2.This author prefers the more general definition given above. The root and the power indicated in a rational exponent can be evaluated in n In l' either order.That is, (ot I )' : (onlt providedat is real.For example,

g2l3:(Btlrz:22:4 or g2l3 : (92)tl3 : 64t13: 4.

A negative rational exponent indicates reciprocal just as a negative integral exponentdoes. So

1 111 ' - - o Jta 6-t- (gt/3)2 22 4'

Evaluate8-213 mefrtally as follows: The cube root of 8 is 2,2 squaredis 4, and the reciprocal of 4 is j. The three operations indicated by a negative rational exponent can be performed in any order, but the simplest procedure for mental evaluationsis summarized as follows.

WWWWWWWWW Evaluatinsd-mtn To evaluatea-'ln mentally, 1. find the nth root of a, 2. raiseit to the m powel 3. find the reciprocal. 28 Chapter P I rffi Prerequisites

Rational exponentscan be reduced to lowest terms. For example, we can evalu- arc 2612by first reducing the exponent: 2612:23:B

Exponents can be reduced only on expressionsthat are real numbers. For example, - t I G g'1' + ( l) because(- 9'1t is not a real number,while ( - I ) is a real number. - ru Your graphing calculatorwill probably evaluate\- t7z1zas 1, becauseit is not using our definition. Moreover, some calculators will not evaluatean expression with a negative base such as (-8;z/:, but will evaluate the equivalent expression ((-8)')1/3.To useyour calculatoreffectively, you must get to know it well. tr

Arc*V/e' I Evatuatingexpressions with rational exponents Evaluate each expression. a. (Z1zlz b. 27-213 c. 1006/a

Solution a. Mentally,the cuberoot of -8 is -2 and the squareof -2 is 4. In symbols: (-21'1t: ((-8)tn)t : (-D2 : 4

-8) b. Mentally,the cube rootof 27 is 3, the squareof 3 is 9, and the reciprocalof 9 is j. { ) In symbols: ( -?/3)rt*rtr, 4 .>t-zl3- I 1-1 L' - - lBE^(6/4) g lEBE (27tlty 32 c. 1006/a: 100312: 103: 1000

wFigure P.20 FE fne expressionsare evaluatedwith a graphing calculator in Fig. P.20. TrV 77ar. Evaluate.a. 9312b. 16-sl4

Rulesfor Rational Exponents The rules for integral exponents from Section P.2 also hold for rational exponents.

Rulesfor The followingrules are valid for all real nunbersa andb andrational numbers RationalExponents r ands, providedthat all indicatedpowers are real and no denominatoris zero. : l. a'e" o'*' 2. { = o'-' 3. (ar)s * qrs a-

/ o\' a' - ( o\-' / t\' o-' 4. (ab),: a,b, t.\;) : u. : ,,. =bt (;/ (;/ b_, a' "

When variable expressionsinvolve even roots, we must be careful with signs. For example, (xz\tlz : * ," not correct for all values of x, because (-51z1ttz : 25t12: 5. Howevequsing absolutevalue we can write

G1112: lxl for everyreal number x P.3r ffiw RatronalExoonents and Radicals 29

When finding an even root of an expressioninvolving variables, remember that if n is even,a'/' islhe positive nthroot of rz. footn//a I usingabsotute value with rational exponents Simplify each expression, using absolute value when necessary.Assume that the variables can representany real numbers. a. (64a\U6 b. Qett/t c. @8Stl+ d. (yt')tlo

Sotution a. For any nonnegativereal number a,wehave (64aa1rle:2a. If a is negative, (64aa1tleis positive and2a is negative.So we write

(6+a\tla : 2ol : 2lol for everyreal number a. b. Foranynonnegativex,wehave(x\tlz: xel3- x3.lf xisnegative,(*e)113 and 13 are both negative. So we have

Qslrlz : *z for every real numberx. c. For nonnegativec, we have @t\t1+ : a2. Since (o')'lo anda2 areboth positiveif a is negative,no absolutevalue sign is needed.So (a\tl4 : a2 for every real numbera. d. Fornonnegativey,wehave(yt')tlo: rz.rcyisnegative, (y'\'lo ispositivebut y3 is negative.So

(yt\tlo : lyt l for every real numbery.

4rV Titl. Let w representany real number. Simplify (r\tb

When simplifying expressionswe often assumethat the variablesrepresent pos- itive real numbers so that we do not have to be concerned about undefined expres- sions or absolute value. In the following example we make that assumption as we use the rules of exponentsto simplify expressionsinvolving rational exponents. fuonF/e I Simplifyingexpressions with rational exponents Use the rules of exponents to simplify each expression.Assume that the variables representpositive real numbers.Write answerswithout negative exponents. ( aztz6ztzlt a. xzl3x4l3 b. @aytlzlrl+ c. tr, \4-./ Sotution

L. x2l3x4l3 : x6/3 Procluctrule _ --2 -L Simplitythe exponcnt. b. (xayllzltl+: G\t14(yll\t14 Power of a productrule

: *!'l' Powero1'a power lule ChapterP rrr Prerequisites

( a3l2b2/3\3 6t/z1z162/t1z : --@- Powerofaquotientrule "' l-7- ) oe1262 Powerofa powerrule u^d lo -;)?\ a-5lLbz QuotientruleV- 6 =

D- 7t) Definition of negativeexponents a't' TrV 1ht. Simpliff(ar/zor/z7rz.

Radical Notation The exponentlf n andtheradical sign V- are both usedto indicatethe nth root.

Definition:Radical

',1 |

The numbera is calledthe radicand andn is the index of the radical.Expressions suchas \/4, {-u, and{/i do not representreal numbersbecause each is an evenroot of a negative.number..

ho*Vle p Evaluatingradicals Evaluateeach expression and cheokwith a calculator. o116 ,. . J1q9b. tf looo t'V81

$olution a. Thesymbol \6 indicatesthe positive squaxe root of 49. So \6 : 49112: 7. Writing\/49 : +7 is incorrect. b. t'f 1000= (-1000)1/3: -10 checkthat(-ro;t: -tooo. olrc (rc\'ro 2 '' : :1 checkthat(3I = V8r \sr/ *f r FigureP.21 Theseexpressions are evaluated with a calculatorin Fig. P.21. TrV Tltt. Evaluate. Vioo b. */-n ".

Sinceall' I t- :, Y^nf a, expressionsinvolving rational exponentscan be written with radicals. P.3r rr RationalExponents and Radicals 31

Rule:Converting atln to RadicalNotation

fua*T/a @ writing rational exponents as radicals Write eachexpression in radicalnotation. Assume that all variablesrepresent posi- tive realnumbers. Simplify the radicandif possible. t Z2l3 b. (3g3la c. 2(x2 + 3)-112

Sotution a. 22/3: t/F : t/+ b. (lx1z/+: {-Arf : tfr*t c. 2(* + 31-t1z (x" + 31'r" t/f + I

TrV \hl. Wirte 5213in radicalnotation. r

The Product and Quotient Rules for Radicals Using rational exponentswe can write

- (ab)r/' o1/n61/n andc)''':# Theseequations say that the nth root ofa product (or quotient)is the product(or quotient) of the zth roots. Using radical notation these rules are written as follows.

Rule:Product and Quotient

An expressionthat is the squareof a term that is free of radicalsis called a perfect square..Forexample, gx6 is a perfectsquaxe because ,*e : (3x3)2.Like- wise,27yr2is a perfect cube. In general,an expressionthat is the nth powerofan expressionfree of radicalsis a perfect zth power. In the next example,the product and quotientrules for radicals are used to simplify radicalscontaining perfect squaxes,cubes, and so on. 32 ChapterPrrr Prerequisites

fuatn7/a- I Uslngtlte product and ryrotient rutes tor radieaLs Simplify eachradical expression.Assume that all variablesrepresent positive real numbers.

^- rr: 1 a. YIZSab b. r/ - c. Y lo

$otrutien a. Both 125 anda6 arcperfectcubes, So use the productrule to simplify:

f/125F : Vns , l/a6 : 5a2 since9F = a6t3= a2

b. Since16 is a perfectsquaxe, use the quotientrule to simpliS'the radical: T: ------::\/t -\/i Y16 t/rc 4

r FigureP.22 ffi We cancheck this answerby usinga calculatoras shown in Fig. P.22,Note that agreementin the first 10 decimalplaces supports our belief that the two ex- pressionsare equal,but doesnot proveit. The expressionsare equalbecause of the quotientrule. tr .l42vs V:rfr -2v c. il-+ : -: : -+ Since\{fr = x2ot5= x4 \ y'" t'/x20 x' Try 77at.simpliff V-8fr

Simplified Form and Rationatizing the Denominator We havebeen simpliffing radical expressionsby just making them look simpler. However,a ra'dicalexpression is in simpliJiedform only if it satisfiesthe following three specific conditions.(You should check that the simplified expressionsof Example7 satisfythese conditions.)

Definition:Simplified Form for Radicalsof lndexn

The productrule is usedto removethe perfectzth powerslhat arefactors of the radicand,and the quotientrule is usedwhen fractionsoccur inside the radical. The processof removingradicals from a denominatoris calledrationalizing the denominator.Radicals can be removedfrom the numeratorby usingthe sametype ofprocedure. P.3r rr RationalExoonents and Radicals 33 foar+7/a- @ Simptified f,orrn and rationalizing the denominator Write eachradical expression in simplifiedform. Assume that all variablesrepresent positivereal numbers. ^. x/n n. t/iW "'+rt'E Sotution a. Since4 is a factorof 20, \/20 is not in its simplifiedform. Usethe productrule for radicalsto simplify it: \/20: \/4. \/t:2\/i b. Use the product rule to factor the radical, putting all perfect squaresin the first factor: Nr#f : \/6F . f6y Productrure

: zrlyol6y Simplifuthe first radical c. Since \6 upp.u., in the denominator,we multiply the numeratorand denqm- inator by t/i to rationalizethe denominator.Note that multiplying-'v3 by # it equivalentto multiplyingthe expressionby 1. So its appeaxanceis changed,but not its value.The following displayillustrates this point. s s s \/t sf, - .-:-:rvr -^^/: '" \/t \/, \/t \/t 3 d. To rationalize this denominator,we must get a perfect cube in the denominator. The radicand5aa can be made into the perfect cube 125a6by multiplying by 25a2: t;- ilr \6 \,1 Quotient rule for radicals 5a4 vi7 \%.\/r* vi7.vri7 Multiply numerator and denominatotby l/Z5o'. ffri? Vn# Product rule for radicals ffiG Since (5a2)3: 125a6 )a- TrV 77a1. write V8l in simplifiedform.

Operationswith RadicalExpressions Radicalexpressions with the sameindex can be adde{ subtracted"multiplied or divided.For example, 2\h + 3\h : 5rt because2x 'f 3x: 5x is true for any value of x. Because 2rt and,3\h are addedin the samemanner as like 34 ChapterP rrffi Prerequisites

terms, they are called like terms or like radicals. Note that sums such as f\ + \/i o, + \/2y be written as a single radical because the "/zy "unnot terms are not like terms. The next example further illustrates the basic operations with radicals.

foa*+7/e p Operationswith radicals of the sameindex Perform each operation and simplifii each answer.Assume that each variable repre- sentsa positive real number. t/n + \/t b. t/24x - \/sr. tW . {W d. {qo * r/i ". ". Solution ".t/n+\/t:\/q.fi+fi Product rule for radicals :2\/t + \/i :3\/t Simplify. Add like terms.

b. V24x- \/8r.: td . $, - \/n . $. Product rule for radicals

:2fl3x-3Vi:-t/k Simplify. Subtract like terms.

t/+f .\/W:'(q8y' Product rule for radicals ". : Factor out the perfect fourth V@.$y: zy{i powers. Simplify. - + : rule for radicals; divide. d.\/40 \/t: V+ \/g Quotient

:\/4.fi:zfi Product rule; simplify.

IrV 7b:. Subtractand simplify f s0 - \/8.

Radicalswith differentindices are not usuallyadded or subtracted,but they can be combinedin certaincases as shownin the next example.

fua*Vle@ Gombining radicats with different indices Write eachexpression using a singleradical symbol. Assume that eachvariable rep- resentsa positivereal number. tf;. . t/i b. t6, . Vry f t6. ". ". Solution ^.rfr..fr:2t/3 . 3112 Rewrite radicals as rational exponents.

22/6. 3316 Write exponentswith the least common denominator.

< /--;---"- y2' .3' Rewrite in radical notation using the product rule.

Vl08 Simpliflz inside the radical. P.3rrr Exercises 35

b. {,. {U: rr/z12r7t/+Rewrite radicalsas rational exponents: : ,+/r214,12/tzWrite exponentswith the LCD. ={fw Rewrite in radical nolation using the productrule. :w Simplifr insidethe radical. \/ t/i : (2rlz1tlz: 2tl6 : g, ". 4fy Tfu:. write \% - \/iusngasingleradicalsymbol.

In Example10(c) we foundthat the squareroot of a cuberoot is a sixth root.In general,an mth roofof an nth root is an mnth rcot.

Theorem:mth Root of an nth Root

For Thought True or False?Explain. Do Not Usea Calculator.

1. 8-1/3: -2F 2. l6U4 : 41/2T 7. grlz: \frp s.+=*, 17. v35 ,.Vi:i' 4. (6)' = 3t/it ,.#:ffi, n. VF :71/+sr 5. FD2/2 = -l F 6. \fr : 7312F

IE Exercises lJsethe procedurefor evaluatinga-'/" on page 27 to anluate 19. (a81rl+oz zo. (zt27t/4151 eachacpression. Use a calculator to check.(Examples I and 2) 21. (fyef/z *rz 22. (l6xay81rl+21r1rz l. -gtP -z Z, 27U3s 3. 64U2e 4. -l44tl2 -t2 Simplifu eachexpression. Assurue that all variablesrepresent pogi- 5. (-6qtl3-4 6, 8l/43 7. (-ZtyoF,, ,, tive real numbers.Write yaur enswerswithout negativeqcponents. !?;i,t @xample4) . y'/3 . s. s-4/3tlrc r0.4-3/2 rls r. ,t, ir. 23. y2/3 f 24. a3/s a1/sd2 0', (*)''t / 1/2 25. (xay)UzSrrtz 26. (au2bu312o6?tt lq\ttz v / 8\2/3 tr. 8t2714. 4/g 27. (2au\(3a) eattz 28. (lyt/t1(Zyt/27 (o/ \-n ) 6rsto

-4Y ,,4 Simpffi eachexpression. Use absolute value whennecessary. zt. /ro. - _1a,llr @xample 3) ffiu'rc 2y'/' 15. (x6)r/61x1 16. (xrolrlsS

17. (ars1t/sot r8. (y\rtz 1r1 36 ChapterP rrffi Prerequisites

\ 4 - ! Perform the indicated operations and simplifu your answer As- 31. la2btt2)lat/rb_t/2)d 32. @314 a2 b3)@3 I a-2 b \ sume that all variables representpositiye real numbers. Whenpos- / .-b-.3 \ l/3 ) sible use a calculator your (Exumple .. l^ y I xy to verifu answer. 9) JJ.t o I r ,0. (!3)' .',','u \z'/ : 77.f8 +f -{izxT + 2\/t - 2\/5 Evaluate each radical expression.Use a calculator to check. - - - (Exumple5) 7s. \48 \,6 + fi \4t zt,5 zfl :s. l6oo:tr za.fqoo2.o 37.V-8 2 ts. (-zx6)(s:,G) 30ft ao.(-:rD)(-zx6)axG q gA2 se.VA 3s.V4 2 40. sl. (3v6X+{sa) eo" n. (-zf e)(zrG)-ze

l4 *r (-srA\z' '") rs'' t+. (zfi)'z+s 4r.\l;213 qz.rf Jttq +r.r,/o.oro.r "*' \ Y It) ^^a 85' Vl8d + 1/2o+::-" s6.f2k1 ={3r'*'Vi .I 8- 0' .v6zso.s - - \\/Y -.,tfi u. 4s..'/ + v , 1000 *il*: 87.5 Yx: 88. a + Vb; - I /5 uso. n. t14ds ts. l/ss zz Bg.\50f + f q5x3s*\,rtu Vt6oo + t'/sql soV2u

Write each expressioninvolving rational exponentsin radical nota- Write each expressionusing a single radical sign. Assume that all tion and each expressioninvolving radicals in exponential notation. variables representpositive real numbers. Simphfy the radicand lExumple6) wherepossible. (ExampleIr,1 49. rc213{Kf 50. _2314 *F fl. 3y-3ls L sr.tA . ft vn s2.\/G . \fr V2ooo 52. a(b+r,+l)-t/2 sz,J- r tt.- 54. _ 4\,G -4x3t') f,6.\%X,Et; vx %. s4.\A.Y4Vn + \,/il | ss.fi. \/2*y{+r',' ' ge.f/zo. 55. {F'3/s s6.\/7;7 (x3+.yr1r/: \6{5r;, gl. 'V t/l tfi sB."V l/2a*A Simplfy each radical expression.Assume that all variables repre- sentpositive real numbers. (Erample7) Solve eachproblem. q' sl. f ta"t s8.f ntya trr' 99. Economic Order Quantity Purchasingmanagers use the formula SS."W z),3 60. \/rzsr8 s," -_ l2AS VI

".ffif ".^l*+ to determine the most economic order quantity E for parts used in production. I is the quantity that the plant will use in u, year, fff- "6a,i@':Y y'.v- one S is the cost of setup for making the part, and 1 is the cost of holding one unit in stock for one year. Find E if S: $6000,A:25,andI: $140.46 Write each radical expressionin simplified form. Assume that all variables representpositive real numbers. (Example8) 100. Piano Tuning The note middle C on a piano is tuned so that L=* the string vibrates at 262 cy cles per second,or 262 Hz 6s.fn2rt 66.xTs$ 6i.++ 4u. (Hertz). The C note that is one octave higher is tuned to Ysr V7 524H2. Tuning for the 11 notes in between using the method a . un. tf*,,',Ft'. of equal temperamentrs 262 2'/t2, wheren takesthe values V; Y' ,o.lX t6zt s 72.tAZ.rr z I through 11. Find the tuning rounded to the nearestwhole Hertz for those 11 notes. I^ 278, 294, 3 t2, 330,3s0, 37 l, 393,41 6, 441,467, 495 Hz B. V-2sor74. V-uo, ,t.#+ !Jx 76.\25 5;r\ 2r - 2u V3i \4t 5 P.3rilm Exercises 37

l0l. Sail Area-DisplacementRatio The sail area-displacement 106. ChangingRaditts The radiusof a spherer is given in ternrs ratio S is given by of its volume Vby the formula

16A ^ : ( o'tsvl'r' r '-\ ;tF. rr ) whereI is the sail area and d is the displacement(lbs). 1ft2.; By how many incheshas the radiusofa sphericalballoon in- S measuresthe amount of power availableto drive a sail- creasedwhen the amountofair in the balloon is increased boat (Ted Brewer Yacht Design, wwwtedbrewer.com). Ra- from 4.2 ft3 to 4.3 ft3? o.ogin. tios typically rangefrom 15 to 25, with a high ratio indicat- ing a powerful boat. Find S for the USS Constitution, which 107. Her"oni;Formttla lf the lengthsof the sidesof a triangleare hasa displacementof 2200 tons,a sail areaof 42,700 ft2, a,b,and c, and.s: (a + b + c)12,then the areal is given and 44 guns.25.4 by the formula 102. A Less PowerfulBoat Find S (from the previous exercise) /_@ for the Ted Hood 5 L It has a sail area of 1302ft2, a dis- placementof 49,400pounds, a length of 5 I feet, and no Find the areaof a trianglewhose sides are 6 ft, 7 ft, and ll ft guns. I_5.5 (seefigure for Exercises107 and 108).19.0 ftr

703, DepreciationRate lfthe cost ofan item is Cand after n 108, Area o/ un Equiluteral Triangle Use Heron's formula yearsits value is,S,then the annualdepreciation rate r is from the previous exerciseto find a formula for the area givenby r : 1 - (SlC)tt'. After a usefullife of r yearsa of an equilateraltriangle with sidesof lengtha and computerwith an original cost of $5000 hasa salvagevalue simplify it. ,,V: of $200. 4 a. Use the accompanyinggraph to estimater if n : 5 and if n : 10.50%,.30,2, ,/ \, b. Usetheformulatofindrifn:5 andifn : 10. 47.5"1'.27 .5"/,, /\

n Figurefor ExercisesfOZana fOe

0.8

* tr.b 'jf For Writing/Discussion '6 0.4 I 109, Rootsor Powers Which onc of the following expressionsis ,i, Q ili O 6.2 not equivalentto the others?Explain in writing how you ar- rived at your decision.b b.VF t/F 246810 ". Usefullif'e (years) ".({,)^ d. ta/s e. (tt/s1t r Figurefor Exercise103 I10. Which oneof the followingexpressions is not equivalentto f ottft Write your reasoningin a paragraph.c 104.BMII/ Depreciation A new BMW Z3 convertiblesells for a. lal .t2 b. lab'zl c. ab2 $30,193while a five-year-oldmodel sells for $17,095 (Edmund's,www.edmunds.com). Use the formula from the d. (a2b\tl2 e. *fu2 previousexercise to find the annualdepreciation rate. l0.l{,2, 7ll. The Lost Rule? Is it true that the squareroot ofa sum is I 05. Longest Screwdriver A toolbox has length I, width Il, and equalto the sum of the squareroots? Explain. Give examples. height H. The length D of the longest screwdriver that will fit No insidethe box is given by ll2. Tbchnicalities lf mandnarereal numbersandm2: n, D: Q2 + W2 + H\t12. then rr is a squareroot of r, but if nr : n then m is the cube Find the length of the longest screwdriver that will fit in a root of r. How do we know when to use "a" or "the"? 4 in. bv 6 in. bv 12in. box. 14in.