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"?