To use your calculator effectively, you must get to know it well. tr. Arc*V/e' I ... r
and s, provided that all indicated powers are real and no denominator is zero. , o-
' ... use the rules of exponents to simplify expressions involving rational
exponents.
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 definitions and rules concerning rational exponentsand radicals.
Roots Since 2a : 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 even or odd"dependingon whetherthe positive integer is 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 number has exactly onereal odd root. For example,because23 : 8 and 3 is odd,2 is the only real cube root 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:Exponent1/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 positive real squareroot of 4. So 4tl2 : 2. b. 8t/3 : 2 c. (_g)t/3: _2
P.3 r ilffi
RationalExoonents and Radicals
27
d. Since the definition of nth root does not include an even root of a negative number, (_471/zhas not yet been defined. Even roots of negativenumbersdo 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. Evaluate. TrVThs.
a. 91/2 b. l6tl4
Rational Exponents We have defined 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: RationalExponents
If m andr arepositiveintegers,then e^ln : @rlr)^ providedthatall' is a realnumber.
Note that a'tn is not real when a is negative and n is even. According to the definition of rationalexponents, expressions suchas (_251-zlz, (*+31t1+,and(- l)21'ate not defined becauseeach ofthem involves an even root ofa nesative number. Note t h a t s o m e a u t h o r sd e f i n e a ' l ' o n l y f o r m f n i n l o w e s t t e r m s .I n - t h a t c a s e t h e f o u r t h 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 either order.That is, ( ot I n)' : ( onlt I n providedat l' 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 o
'
-
1 Jta 6-t-
-
111 (gt/3)2
22
4'
Evaluate8-213mefrtally 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
E v a l u a t idn-sm t n 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
Prerequ isites Rational exponentscan be reduced to lowest terms. For example, we can evaluarc 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. - 1, becauseit is graphing probably Your calculator will evaluate t7z1z as \ru 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
Evatuating expressions 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: ((-8)tn)t :
(-21'1t:
{ -8)
)
( -?/3)rt*rtr, 4
lBE^(6/4)
b. Mentally,the cube rootof 27 is 3, the squareof 3 is 9, and the reciprocalof 9 is j. In symbols: .>t-zl3L'
lEBE c. 1006/a:
wFigureP.20
(-D2 : 4
FE
100312: 103 :
I - 1 - -1 (27tlty 32 g
1000
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 RationalExponents
The followingrulesarevalid for all real nunbersa andb andrationalnumbers r ands, providedthatall indicatedpowersarerealandno denominator is zero. l. a'e" :
o'*'
2. { = o'-' a/ o\'
4. (ab),: a,b, t.\;)
:
a'
3. (ar)s * -
( o\-'
u. (;/
qrs
:
/ t\'
(;/
,, . o - ' = b t 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 everyrealnumberx
P.3r ffiw
Ratronal Exoonents and Radicals
29
When finding an even root of an expressioninvolving variables, remember that if n is even,a'/' islhe positive nthroot of rz.
usingabsotutevaluewith
footn//a I
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. (64aa1tleis positive and2a is negative.So we write (6+a\tla :
2ol
: 2lol
If a is negative,
for everyreal numbera.
b . F o r a n y n o n n e g a t i v e x , w e h a v e ( x \ t l z : x e l 3 - x 3 . l f x i s n e g a t i v e (, * e ) 1 1a3n d 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 number a.
d. Fornonnegativey,wehave(yt')tlo: y3 is negative.So (yt\tlo 4rV
:
lyt l
rz.rcyisnegative, (y'\'lo ispositivebut for every real numbery.
Let w representany real number. Simplify (r\tb
Titl.
When simplifying expressionswe often assumethat the variablesrepresentpositive real numbers so that we do not have to be concerned about undefined expressions or absolute value. In the following example we make that assumption as we use the rules of exponentsto simplify expressionsinvolving rational exponents.
Simplifyingexpressions with
fuonF/e I
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. t r , \4-./
Sotution L. x2l3x4l3 :
x6/3
_ --2 -L
b. (xayllzltl+:
Procluctrule S i m p l i t yt h e e x p o n c n t .
G\t14(yll\t14
: *!'l'
Power of a product rule P o w e ro 1 ' ap o w e rl u l e
ChapterP rrr
Prerequisites ( a3l2b2/3\3 6t/z1z162/t1z : --@Powerofaquotientrule "' l-7) oe1262
Powerofa powerrule
u^d a-5lLbz
lo -;)?\ QuotientruleV- 6 =
D7t) a't'
Definition of negativeexponents
TrV 1ht. Simpliff(ar/zor/z7rz.
Radical Notation The exponentlf n andtheradical sign V-
areboth usedto indicatethe nth root.
Definition:Radical
',1
|
The numbera is calledthe radicand andn is the index of the radical.Expressions and{/i do not representreal numbersbecauseeachis an suchas \/4, {-u, evenroot of a negative.number. .
ho*Vle
p
Evaluatingradicals
Evaluateeachexpressionand cheokwith a calculator.
,. . J1q9b. tf
o116 looo t ' V 8 1
$olution thepositivesquaxe rootof 49.So \6 a. Thesymbol\6 indicates +7 : incorrect. is Writing\/49 : -10 checkthat (-ro;t: -tooo. b. t'f 1000= (-1000)1/3
olrc: (rc\'ro :12 '' V8r \sr/ r FigureP.21
: 49112 : 7.
checkthat (3I = *f
areevaluatedwith a calculatorin Fig. P.21. Theseexpressions
TrV Tltt. Evaluate. Vioo b. */-n ". I t^nf Sinceall' :, Y a, expressionsinvolving rational exponentscanbe written with radicals.
P.3 r rr
RationalExponents and Radicals
31
Rule:Convertingatln to RadicalNotation
fua*T/a
@ writing rationalexponents as radicals
Write eachexpressionin radicalnotation.Assumethat all variablesrepresentpositive realnumbers.Simplifythe 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 TrV \hl.
(x" + 31'r"
t/f
+I
Wirte 5213 in radicalnotation.
r
The Product and Quotient Rules for Radicals Using rational exponentswe canwrite (ab)r/' -
o1/n61/n
andc)''':#
Theseequationssaythat 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 calleda perfect square..Forexample,gx6is a perfectsquaxebecause,*e : (3x3)2.Likewise,27yr2is a perfect cube.In general,an expression that 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 radicalscontainingperfect squaxes, cubes,andso on.
32
ChapterPrrr
Prerequisites
Uslngtlte productand ryrotient rutes tor radieaLs
fuatn7/a I -
Simplify eachradical expression. Assumethat all variablesrepresentpositivereal numbers. ^-
a. YIZSab
r:
r
1
b. r/ -
Y lo
c.
$otrutien a. Both 125 anda6 arcperfectcubes,Sousetheproductrule to simplify: , l/a6 : 5a2 since9F = a6t3 = a2
f/125F : Vns
b. Since16 is a perfectsquaxe, usethe quotientrule to simpliS'the radical:
T: Y16 r FigureP.22
\/t
\/i
t/rc
4
------:: -
We cancheckthis answerby usinga calculatorasshownin Fig. P.22,Note ffi that agreement in the first 10 decimalplacessupportsour belief that the two expressionsare equal,but doesnot proveit. The expressions are equalbecauseof the quotientrule. tr
.l42vs: -:V:rfr
c. il-+ \ y'"
-2v
: -+ x' t'/x20
= x4 Since\{fr = x2ot5
Try 77at.simpliffV-8fr Simplified Form and Rationatizing the Denominator We havebeen simpliffing radical expressions by just making them look simpler. However,a ra'dicalexpressionis in simpliJiedform only if it satisfiesthe following three specific conditions.(You should check that the simplified expressionsof Example7 satisfytheseconditions.)
Definition:SimplifiedForm for Radicals of lndexn
The productrule is usedto removethe perfectzth powerslhat arefactorsof the radicand,andthe quotientrule is usedwhen fractionsoccurinsidethe radical. The processof removingradicalsfrom a denominatoris calledrationalizing the denominator.Radicalscanbe removedfrom the numeratorby usingthe sametype ofprocedure.
P.3r rr
foar+7/a -
RationalExoonents and Radicals
33
@ Simptified f,orrn and rationalizing the denominator
Write eachradicalexpression in simplifiedform.Assumethat all variablesrepresent positiverealnumbers.
^. 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
c. Since \6
: \/6F . f6y : zrlyol6y
Productrure Simplifuthe first radical
upp.u., in the denominator,we multiply the numeratorand denqm-
inator by t/i
to rationalizethe denominator.Note that multiplying - ' vby3 #
it
equivalentto multiplyingthe expressionby 1. So its appeaxance is changed,but not its value.The following displayillustratesthis point.
s
\/t
s -
\/,
s
\/t
. - : - : r v r
\/t \/t
sf,
-^^/: '"
3
d. To rationalizethis 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 \,1 5a4
\6
vi7 \%.\/r*
vi7.vri7 ffri? Vn# ffiG )a-
TrV 77a1. write V8l
Quotient rule for radicals
Multiply numerator and denominatotby l/Z5o'.
Product rule for radicals
Since (5a2)3 :
125a6
in simplifiedform.
Operationswith RadicalExpressions Radicalexpressionswith the sameindex can be adde{ subtracted" multiplied or 'f : divided.For example 5rt because2x 3x: 5x is true for , 2\h + 3\h any value of x. Becaus e 2rt and,3\h are addedin the samemanneras like
34
ChapterP rrffi
Prerequisites terms, they are called like terms or like radicals. Note that sums such as + \/i o, + \/2y f\ 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 same index
Perform each operation and simplifii each answer.Assume that each variable representsa positive real number.
".
t/n + \/t
b. t/24x- \/sr.
".
tW . {W
d. {qo * r/i
Solution
".t/n+\/t:\/q.fi+fi :2\/t + \/i :3\/t b. V24x- \/8r.: td . $, - \/n . $. :2fl3x-3Vi:-t/k t/+f .\/W:'(q8y' ". : V@.$y: zy{i
Product rule for radicals Simplify. Add like terms. Product rule for radicals Simplify. Subtract like terms. Product rule for radicals Factor out the perfect fourth powers. Simplify.
-
d.\/40+ \/t: V+ : \/g
Quotient rule for radicals; divide.
:\/4.fi:zfi
Product rule; simplify.
IrV 7b:. Subtract andsimplifyf s0 - \/8. Radicalswith differentindicesarenot usuallyaddedor subtracted, but they can be combinedin certaincasesasshownin thenextexample.
fua*Vle@ Gombining radicats with different indices Write eachexpression usinga singleradicalsymbol.Assumethat eachvariablerepresentsa positiverealnumber.
".
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
b. {,.
{U:
Exercises 3 5
rr/z12r7t/+Rewriteradicalsasrational exponents: : ,+/r214,12/tz Write exponentswith the LCD.
={fw
:w
Rewrite in radical nolation using the productrule. Simplifr insidethe radical.
\/ t/i : (2rlz1tlz: 2tl6 : g, ". 4fy Tfu:. write \% - \/iusngasingleradicalsymbol. In Example10(c)we foundthatthe 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
17.
,.Vi:i'
4. (6)'
5. FD2/2 = -l F
6. \fr : 7312 F
IE
7. grlz: \frp
= 3t/it
,.#:ffi,
l. -gtP -z
Z, 27U3s
5. (-6qtl3-4 6, 8l/43
3. 64U2e
4. -l44tl2 -t2
7. (-ZtyoF ,,
,,
s. s-4/3 tlrc r0.4-3/2 rls r. lq\ttz
n. VF :71/+sr
Exercises
lJsetheprocedurefor evaluatinga-'/" onpage 27 to anluate eachacpression.Usea calculator to check.(Examples I and2)
tr. (o/
s.+=*, v35
v
0',
,t, ir. (*)''t
/ /
8\2/3
8t2714. \-n )
!?;i,t 1/2
4/g
Simpffi eachexpression.Useabsolutevalue whennecessary. @xample 3)
15. (x6)r/61x1
16. (xrolrlsS
17. (ars1t/s ot
r8. (y\rtz 1r1
19. (a81rl+ oz
zo. (zt27t/4 151
21. (fyef/z *rz
22. (l6xay81rl+ 21r1rz
Simplifueachexpression.Assuruethat all variablesrepresentpogitive real numbers.Writeyaur enswerswithout negativeqcponents. 4) @xample 23. y2/3 . y'/3 f 24. a3/s. a1/sd2 25. (xay)UzSrrtz
26. (au2bu312 o6?tt
27. (2au\(3a) eattz
28. (lyt/t1(Zyt/27
zt.ffiu'rc
/ro. - 4 Y -
2y'/'
,,4
_1a,llr
6rsto
ChapterP rrffi
36
3 1 . l a 2 b t t 2 ) l a t / r b _ t / 2d) ..
JJ.t
/ .-b-.3 \ l^ y I o
l/3
Prerequisites 32. @314a2 b3)@3I4a-2 b \ !
\
)
xy
I
:
,0.(!3)' .',','u
r
Perform the indicated operations and simplifu your answer Assume that all variables representpositiye real numbers. Whenpossible use a calculator to verifu your answer.(Exumple9)
Evaluate each radical expression.Use a calculator to check. (Exumple5)
-{izxT + 2\/t - 2\/5 77.f8 +f 7s.\48 - \,6 + fi - \4t zt,5- zfl
:s. l6oo:tr
ts. (-zx6)(s:,G) 30ft
\z'/
za. fqoo2.o
se.VA q
2
3s.V4
l4
4 r .\ l ; 2 1 3
qz.rfJttq Y It)
u. .v6zso.s
.I 8 4s..'/v , 1000 -
n. t14ds
ts. l/ss zz
37.V-8
2
40.gA2
sl. (3v6X+{sa) eo"
ao.(-:rD)(-zx6)axG n. (-zf e)(zrG)-ze
+r. r,/o .o ro.r
*" *r ' (\ - s' r A ' " \) z r' 's
t+. (zfi)'z+s
85' Vl8d
*il*:
8 7 .5 + Y x :
Bg.\50f
Write each expressioninvolving rational exponentsin radical notation and each expressioninvolving radicals in exponential notation. lExumple6) 49. rc213{Kf 50. _2314 *F fl. 3y-3ls L 5 2 . a ( b + r , +l ) - t / 2
\,/il + | 55. {F'3/s
s6. \/7;7
54. _ 4\,G -4x3t')
-.,tfi
\ \ / Y
88.a + Vb;
uso. + f q5x3 s*\,rtu Vt6oo + t'/sql soV2u
Write each expressionusing a single radical sign. Assume that all variables representpositive real numbers. Simphfy the radicand wherepossible. (ExampleIr,1
sr.tA . ft vn
s2.\/G . \fr V2ooo
%. f,6.\%X,Et; ss.fi.
(x3+.yr1r/:
s 6 .f 2 k 1 = { 3 r ' * ' V i
0'
-
I /5
sz,J- r tt.vx
^^a + 1/2o+::-"
s 4 .\ A . Y 4 V n ' ge.f/zo. \6{5r;,
\/2*y{+r','
gl. 'V t/l tfi Simplfy each radical expression.Assume that all variables representpositive real numbers. (Erample7)
sl. f ta"t q' SS."W
s 8 .f n t y a t r r '
z),3
".^l*+
u,fff-
" 6 a Y, iy@ ' . v -' :
Write each radical expressionin simplified form. Assume that all variables representpositive real numbers. (Example8)
6s. fn2rt
66. xTs$ 6i.++ Ysr
4u.L=* V7
a
tf*,, un.V; Y' ,o.lX ',Ft'. t6zt s 72.tAZ.rr z
B. V-2sor74.V-uo, ,t.#+ 5;r\ 2r
- 2u V3i
Solve eachproblem. 99. Economic Order Quantity formula
60. \/rzsr8 s,"
".ffif
sB."Vl/2a*A
I^
!Jx 76. \25
\4t 5
-_
Purchasingmanagersuse the
l2AS VI
to determine the most economic order quantity E for parts used in production. I is the quantity that the plant will use in one year, 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 : $ 6 0 0 0A, : 2 5 , a n d I : $140.46 100. Piano Tuning The note middle C on a piano is tuned so that the string vibrates at 262 cy cles per second,or 262 Hz (Hertz). The C note that is one octave higher is tuned to 524H2. Tuning for the 11 notes in between using the method of equal temperamentrs 262 . 2'/t2, wheren takesthe values I through 11. Find the tuning rounded to the nearestwhole Hertz for those 11 notes. 278, 294,3 t2, 330,3s0,37l, 393,416, 441,467,495Hz
P.3 rilm l0l.
Sail Area-DisplacementRatio ratio S is given by ^ r :
The sail area-displacement
16A '-\
b . U s e t h e f o r m u l a t o f i n d r i nf : 5 47.5"1'.27 .5"/,,
andifn :
( o'tsvl'r' rr )
By how many incheshas the radius ofa sphericalballoon increasedwhen the amountofair in the balloon is increased from 4.2 ft3 to 4.3 ft3? o.ogin. 107. Her"oni;Formttla lf the lengthsof the sidesof a triangleare a,b,and c, and.s : (a + b + c)12,then the areal is given by the formula
/_@
102. A Less Powerful Boat Find S (from the previous exercise) for the Ted Hood 5 L It has a sail area of 1302ft2, a displacementof 49,400pounds, a length of 5 I feet, and no guns. I _5.5 703, DepreciationRate lfthe cost ofan item is Cand after n yearsits value is,S,then the annualdepreciationrate r is givenby r : 1 - (SlC)tt'. After a usefullife of r yearsa computerwith an original cost of $5000 has a salvagevalue of $200. a. Use the accompanyinggraph to estimater if n : 5 and if n : 10.50%,. 30,2,
37
106. ChangingRaditts The radius of a spherer is given in ternrs of its volume Vby the formula
;tF.
whereI is the sail area1ft2.;and d is the displacement(lbs). S measuresthe amount of power available to drive a sailboat (Ted Brewer Yacht Design, wwwtedbrewer.com). Ratios typically range from 15 to 25, with a high ratio indicating a powerful boat. Find S for the USS Constitution, which has a displacementof 2200 tons, a sail areaof 42,700 ft2, and 44 guns. 25.4
Exercises
Find the areaof a triangle whosesidesare 6 ft, 7 ft, and ll ft (seefigure for Exercises107and 108). 19.0ftr 108, Area o/ un Equiluteral Triangle Use Heron's formula from the previous exerciseto find a formula for the area of an equilateraltrianglewith sidesof lengtha and simplify it. ,,V: 4
,/
10.
\,
/\ n Figurefor Exercises f O Za n a f O e
0.8 * tr.b
For Writing/Discussion
'jf
'6 0.4 I ,i,
109, Rootsor Powers Which onc of the following expressionsis not equivalentto the others?Explain in writing how you arrived at your decision.b
Q
ili O 6.2
246810 Usefullif'e(years)
".({,)^ d. ta/s
r Figurefor Exercise103
7ll.
D:
Q2 + W2+ H\t12.
Find the length of the longest screwdriver that will fit in a 4 in. bv 6 in. bv 12 in. box. 14in.
e. (tt/s1t
f ottft Write your reasoningin a paragraph.c
$30,193while a five-year-oldmodel sellsfor $17,095 (Edmund's,www.edmunds.com).Use the formula from the previousexerciseto find the annualdepreciationrate. l0.l{,2,
height H. The length D of the longest screwdriver that will fit insidethe box is given by
". t/F
I10. Which one of the followingexpressions is not equivalentto
104.BMII/ Depreciation A new BMW Z3 convertiblesells for
I 05. Longest Screwdriver A toolbox has length I, width Il, and
b. VF
a. lal .t2
b. lab'zl
d. (a2b\tl2
e. *fu2
c. ab2
The Lost Rule? Is it true that the squareroot ofa sum is equalto the sum of the squareroots?Explain.Give examples. No
ll2. Tbchnicalities lf mandnarereal numbersandm2: n, then rr is a squareroot of r, but if nr : n then m is the cube root of r. How do we know when to use "a" or "the"?
