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.