Embed Size (px)
Citation preview
26 Chapter P r rn Prerequisites
Rational Exponents and Radicals
Definition: nth Roots
Definition: Exponent 1 /n
lf n is a positive integer and an : b, then a is called an nth root of b. Ifa2 : b, then a is a square root ofD. lf a3 : b,thena is the cube root ofb.
Raising a number to a power is reversed by finding the root of a number. we indicateroots by using rational exponents or radicals. In this section we will review defini-tions and rules concerning rational exponents and radicals.
RootsSince 2a : 16 and (-D4 : 16, both 2 and -2 are fourth roots of 16. The nth rootof a number is defined in terms of the nth Dower.
we also describe roots as even or odd" depending on whether the positive integer iseven or odd. For example, iffl is even (or odd) and a is an nth root ofb, then a iscalled an even (or odd) root of b. Every positive real number has two real evenroots, a positive root and a negative root. For example, both 5 and -5 are squareroots of 25 because 52 :25 and (-5)2 : 25. Moreover, every real number hasexactly onereal odd root. For example, because 23 : 8 and 3 is odd,2 is the onlyreal cube root of 8. Because (-2)3 : -8 and 3 is odd, -2 is the only real cuberoot of -8.
Finding an nth root is the reverse of finding an nth power, so we use the notationat/n for the nth root of a. For example, since the positive square root of 25 is 5, wewrite25t/2 : 5.
If z is a positive even integer and a is positive, then at/n denotes the positivereal 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 denotes the realnth root ofa.
If n is a positive integer, then 0lln : 0.
fua.*V/e I rvaluating expressions involvingexponent 1/n
Evaluate each expression.
a. 4t l2 b. gr /3 c . ( -g; t / : d . (+yr lz
Solution
a. The expression 4ll2 represents the positive real square root of 4. So 4tl2 : 2.b. 8t/3 : 2c . ( _g ) t / 3 : _2
P.3 r ilffi Rational Exoonents and Radicals 27
d. Since the definition of nth root does not include an even root of a negative num-ber, (_471/z has not yet been defined. Even roots of negative numbers do exist inthe complex number system, which we define in Section P.4. So an even root of anegative number is not a real number.
Evaluate. a. 91/2 b. l6tl4
Rational ExponentsWe have defined otl" asthe nthroot of a. We extend this definitionto e'ln,which isdefined as the mth power of the nth root of a. A rational exponent indicates both aroot and a power.
TrVThs.
Definit ion:Rational Exponents
WWWWWWWWW Evaluat ins d-mtn
If m and r are positive integers, then
e^ln : @rlr)^
provided that all' is a real number.
Note that a'tn is not real when a is negative and n is even. According to the defini-t ion of rational exponents, expressions such as (_251-zlz, (*+31t1+,and (- l)21'atenot defined because each ofthem involves an even root ofa nesative number. Notethatsomeauthors def inea' l ' on ly for mfnin lowest terms. In- thatcasethefour thpower of the square root of 3 could not be written as 3a/2. This author prefers themore general definition given above.
The root and the power indicated in a rational exponent can be evaluated ineither order. That is, ( ot I n
)' : ( onlt I n provided at l' is real. For example,
g 2 l 3 : ( B t l r z : 2 2 : 4 o r g2l3 : (92)t l3 : 64t13 : 4.
A negative rational exponent indicates reciprocal just as a negative integralexponent does. So
1o ' -
J t a -
6 - t -
Evaluate 8-213 mefrtally as follows: The cube root of 8 is 2,2 squared is 4, and thereciprocal of 4 is j. The three operations indicated by a negative rational exponentcan be performed in any order, but the simplest procedure for mental evaluations issummarized as follows.
1 1 1(g t /3 )2 22 4 '
To evaluate a-'ln mentally,
1. find the nth root of a,2. raise it to the m powel
3. f ind the reciprocal.
28 Chapter P I rff i Prerequ isites
w Figure P.20
Rules forRational Exponents
Rational exponents can be reduced to lowest terms. For example, we can evalu-arc 2612 by first reducing the exponent:
2 6 1 2 : 2 3 : B
Exponents can be reduced only on expressions that are real numbers. For example,
G g'1' + ( - l) t because (- 9'1t is not a real number, while ( - I ) I is a real number.
ru Your graphing calculator will probably evaluate \- t7z1z as - 1, because it isnot using our definition. Moreover, some calculators will not evaluate an expressionwith a negative base such as (-8;z/:, but will evaluate the equivalent expression((-8)')1/3.To use your calculator effectively, you must get to know it well. tr
Arc*V/e' I Evatuating expressions withrational exponents
Evaluate each expression.
a. (Z1zlz b. 27-213 c. 1006/a
Solution
a. Mentally, the cube root of -8 is -2 and the square of -2 is 4. In symbols:
( -21 '1 t : ( ( - 8 ) t n ) t : ( -D2 : 4
b. Mentally, the cube rootof 27 is 3, the square of 3 is 9, and the reciprocal of 9 is j.
In symbols:
. > t - z l 3 - I - 1 - 1L '
( 27 t l t y 32 -
g
c. 1006/a : 100312: 103 : 1000
FE fne expressions are evaluated with a graphing calculator in Fig. P.20.
TrV 77ar. Evaluate. a. 9312 b. 16-sl4
Rules for Rational ExponentsThe rules for integral exponents from Section P.2 also hold for rational exponents.
The following rules are valid for all real nunbers a and b and rational numbersr and s, provided that all indicated powers are real and no denominator is zero.
, o - ', .
b _ ,
When variable expressions involve even roots, we must be careful with signs.For example, (xz\tlz : * ," not correct for all values of x, because(-51z1ttz : 25t12 : 5. Howeveq using absolute value we can write
l . a'e" : o'* ' 2. { = o'- ' 3. (ar)s * qrsa-
/ o \ ' a ' - ( o \ - ' / t \ '4. (ab), : a,b, t . \ ; ) :
"
u. ( ; / :
( ; /= b t
a '
{ -8 ) )4( -?/3) rt*rtr,
lBE^(6/4)lEBE
G1112 : lxl for every real number x
P.3 r f f iw Ratronal Exoonents and Radicals 29
When finding an even root of an expression involving variables, remember that if nis even, a'/' islhe positive nthroot of rz.
footn//a I using absotute value withrational exponents
Simplify each expression, using absolute value when necessary. Assume that thevariables can represent any real numbers.
a. (64a\U6 b. Qett/t c. @8Stl+ d. (yt')t lo
Sotutiona. For any nonnegative real number a,wehave (64aa1rle :2a. If a is negative,
(64aa1tle is positive and2a is negative. So we write
(6+a\tla : 2ol : 2lol for every real number a.
b. Foranynonnegat ivex,wehave(x\ t lz : xe l3 - x3. l f x isnegat ive, (*e)113 and13 are both negative. So we have
Qslrlz : *z for every real numberx.
c. For nonnegative c, we have @t\t1+ : a2. Since (o')'lo and a2 are both positive ifa is negative, no absolute value sign is needed. So
(a\tl4 : a2 for every real number a.
d. Fornonnegativey,wehave (yt')t lo: rz.rcyisnegative, (y'\ ' lo ispositivebuty3 is negative. So
(yt\tlo : lyt l for every real numbery.
4rV Titl. Let w represent any real number. Simplify (r\tb
When simplifying expressions we often assume that the variables represent 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 weuse the rules of exponents to simplify expressions involving rational exponents.
fuonF/e I Simplifying expressions withrational exponents
Use the rules of exponents to simplify each expression. Assume that the variablesrepresent positive real numbers. Write answers without negative exponents.
( aztz6ztzltt r ,\ 4 - . /
Procluct rule
Simpl i ty the exponcnt.
Power of a product rule
Power o1'a power lu le
a. xzl3x4l3 b. @aytlzlrl+ c.
Sotution
L. x2l3x4l3 : x6/3
_ --2- L
b. (xayllzlt l+ : G\t14(yll\t14
: *! ' l '
Chapter P rrr Prerequisites
Power ofa power rule
lo ?\
QuotientruleV- 6 = -;)
Defi nition of negative exponents
TrV 1ht. Simpliff (ar/zor/z7rz.
Definition: Radical
r Figure P.21
( a3l2b2/3\3 6t/z1z 162/t1z
"' l-7- ) : --@- Powerofaquotientrule
oe1262^du
a-5lLbz
D-7 t )
a ' t '
Radical NotationThe exponent lf n andthe radical sign V- are both used to indicate the nth root.
' , 1 |
The number a is called the radicand and n is the index of the radical. Expressions
such as \/4, {-u, and {/i do not represent real numbers because each is aneven root of a negative.number. .
ho*Vle p Evaluating radicalsEvaluate each expression and cheok with a calculator.
,. . J1q9 b. tf looo o116t 'V81
$olutiona. The symbol \6 indicates the 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' ' V8r : \sr / :1 checkthat (3I = *f
These expressions are evaluated with a calculator in Fig. P.21.
TrV Tltt. Evaluate. ".
Vioo b. */-n
I t- ^nfSince all' :, Y a, expressions involving rational exponents can be written withradicals.
Rule: Converting atln toRadical Notation
Rule: Productand Quotient
P.3 r rr Rational Exponents and Radicals 31
fua*T/a @ writing rational exponentsas radicals
Write each expression in radical notation. Assume that all variables represent posi-tive real numbers. Simplify the radicand if 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 5213 in radical notation. r
The Product and Quotient Rules for RadicalsUsing rational exponents we can write
(ab)r/' - o1/n61/n and c)''':#These equations say that the nth root ofa product (or quotient) is the product (orquotient) of the zth roots. Using radical notation these rules are written asfollows.
An expression that is the square of a term that is free of radicals is called aperfect square..For example, gx6 is a perfect squaxe because ,*e : (3x3)2. Like-wise,27yr2 is a perfect cube. In general, an expression that is the nth power ofanexpression free of radicals is a perfect zth power. In the next example, the productand quotient rules for radicals are used to simplify radicals containing perfectsquaxes, cubes, and so on.
32 ChapterP rrr Prerequisites
r Figure P.22
Definition: Simplified Formfor Radicals of lndex n
fuatn7/a I Uslng tlte product and ryrotient rutes- tor radieaLs
Simplify each radical expression. Assume that all variables represent positive realnumbers.
r:^ - r 1
a. YIZSab b. r / - c .Y l o
$otrutien
a. Both 125 and a6 arcperfect cubes, So use the product rule to simplify:
f/125F : Vns , l/a6 : 5a2 since 9F = a6t3 = a2
b. Since 16 is a perfect squaxe, use the quotient rule to simpliS' the radical:
T:
Y 1 6\/t \/i------:: -t/rc 4
ffi We can check this answer by using a calculator as shown in Fig. P.22, Notethat agreement in the first 10 decimal places supports our belief that the two ex-pressions are equal, but does not prove it. The expressions are equal because ofthe quotient rule. tr
.l42vs V:rfr -2vc. il-+ : -: : -+ Since \{fr = x2ot5 = x4\ y'" t'/x20 x'
Try 77at. simpliff V-8fr
Simplified Form and Rationatizingthe DenominatorWe have been simpliffing radical expressions by just making them look simpler.However, a ra'dical expression is in simpliJied form only if it satisfies the followingthree specific conditions. (You should check that the simplified expressions ofExample 7 satisfy these conditions.)
The product rule is used to remove the perfect zth powers lhat are factors ofthe radicand, and the quotient rule is used when fractions occur inside the radical.The process of removing radicals from a denominator is called rationalizing thedenominator. Radicals can be removed from the numerator by using the same typeofprocedure.
P.3 r rr Rational Exoonents and Radicals 33
foar+7/a @ Simptified f,orrn and rationalizing- the denominator
Write each radical expression in simplified form. Assume that all variables representpositive real numbers.
^. x/n n. t/iW
Sotutiona. Since 4 is a factor of 20, \/20 is not in its simplified form. Use the product rule
for radicals to simplify it:
\ /20: \ /4 . \ / t :2 \ / i
b. Use the product rule to factor the radical, putting all perfect squares in the firstfactor:
Nr#f : \/6F . f6y Productrure
: zrlyol6y Simplifu the first radical
c. Since \6 upp.u., in the denominator, we multiply the numerator and denqm-
inator by t/i to rationalize the denominator. Note that multiplying by # it- ' v 3equivalent to multiplying the expression by 1. So its appeaxance is changed, butnot its value. The following display illustrates this point.
s s s \/t sf, ^^/:. - : - : r v r
\/t \/, - \/t \/t 3 - '"
d. To rationalize this denominator, we must get a perfect cube in the denominator.The radicand 5aa can be made into the perfect cube 125a6 by multiplyingby 25a2:
"'+rt'E
t;-i l r\,1 5a4
\6vi7\%.\/r*vi7.vri7ffri?Vn#ffiG
Quotient rule for radicals
Multiply numerator and denominatotby l/Z5o'.
Product rule for radicals
Since (5a2)3 : 125a6)a-
TrV 77a1. write V8l in simplified form.
Operations with Radical ExpressionsRadical expressions with the same index can be adde{ subtracted" multiplied ordivided. For example , 2\h + 3\h : 5rt because 2x 'f 3x: 5x is true forany value of x. Becaus e 2rt and, 3\h are added in the same manner as like
34 ChapterP rr f f i Prerequisi tes
terms, they are called like terms or like radicals. Note that sums such as
f\ + \/i o, "/zy + \/2y "unnot
be written as a single radical because theterms are not like terms. The next example further illustrates the basic operationswith radicals.
foa*+7/e p Operations with radicalsof the same index
Perform each operation and simplifii each answer. Assume that each variable repre-sents a positive real number.
". t/n + \/t b. t/24x - \/sr.
". tW . {W d. {qo * r/i
Solution
" . t /n+\ / t : \ /q . f i+ f i:2\/t + \/ i :3\/tb. V24x - \/8r.: td . $, - \/n . $.
:2 f l3x-3Vi : - t /k
". t /+f .\ /W:'(q8y'
: V@.$y: zy{ i-
d. \/40 + \/t: V+ : \/g: \ /4 . f i :z f i
IrV 7b:. Subtract and simplify f s0 - \/8.
Product rule for radicals
Simplify. Add like terms.
Product rule for radicals
Simplify. Subtract like terms.
Product rule for radicals
Factor out the perfect fourthpowers. Simplify.
Quotient rule for radicals; divide.
Product rule; simplify.
Radicals with different indices are not usually added or subtracted, but they canbe combined in certain cases as shown in the next example.
fua*Vle@ Gombining radicats withdifferent indices
Write each expression using a single radical symbol. Assume that each variable rep-resents a positive real number.
". tf;. . t/i b. t6, . Vry ". f t6.
Solution
^.rfr..fr:2t/3 . 3112
22/6 . 3316< /--;---"-y2 ' . 3 '
Vl08
Rewrite radicals as rational exponents.
Write exponents with the least common denominator.
Rewrite in radical notation using the product rule.
Simpliflz inside the radical.
P.3 rrr Exercises
Rewrite radicals as rational exponents:
Write exponents with the LCD.
Rewrite in radical nolation using the product rule.
Simplifr inside the radical.
35
b. {,. {U: rr/z12r7t/+: ,+/r214,12/tz={fw:w
". \/ t/i : (2rlz1tlz : 2tl6 : g,
4fy Tfu:. write \% - \/iusngasingleradicalsymbol.
In Example 10(c) we found that the square root of a cube root is a sixth root. Ingeneral, an mth roofof an nth root is an mnth rcot.
Theorem: mth Rootof an nth Root
7. grlz : \frp
,.#:ffi,
s.+=*,v35
n. VF :71/+sr
Simplifu each expression. Assurue that all variables represent pogi-tive real numbers. Write yaur enswers without negative qcponents.@xample 4)23. y2/3 . y'/3 f 24. a3/s . a1/s d2
28.
/ro.
15. (x6)r/6 1x1
17. (ars1t/s ot
16. (xrolrls S
r8. (y\rtz 1r1
For ThoughtTrue or False? Explain.
1. 8-1/3 : -2F
17.,.Vi:i'5. FD2/2 = - l F
Do Not Use a Calculator.
2. l6U4 : 41/2 T
4. (6)' = 3t/it
6. \fr : 7312 F
4. - l44tl2-t2
IE ExerciseslJse the procedurefor evaluating a-'/" on page 27 to anluateeach acpression. Use a calculator to check. (Examples I and 2)
l. -gtP -z Z, 27U3 s 3. 64U2 e
5. (-6qtl3 -4 6, 8l/4 3 7. (-ZtyoF ,, ,, !?;i,t
s. s-4/3 tlrc r0. 4-3/2 rls r. 0',
,t, ir. (*)''t/ 1 /2
lq\ttz v / 8\2/3tr. (o/ 8t27 14. \-n )
4/g
Simpffi each expression. Use absolute value when necessary.@xample 3)
19. (a81rl+ oz
21. (fyef/z *rz
25. (xay)Uz Srrtz
27. (2au\(3a) eattz
zt. ffiu'rc
zo. (zt27t/4 151
22. (l6xay81rl+ 21r1rz
26. (au2bu312 o6?tt
(lyt/t1(Zyt/27 6rsto
- 4Y , , 4- _ 1 a , l l r
2y'/'
36 Chap te rP r r f f i Prerequisites
32. @3 14 a2 b3) @3 I 4 a-2 b
- \ !
,0. (!3)' .',','u
3 1 . l a 2 b t t 2 ) l a t / r b _ t / 2 ) d \
/ . - b - . 3 \ l / 3 ). . l ^ y I x yJ J . t o I r
\ z ' / :
Perform the indicated operations and simplifu your answer As-sume that all variables represent positiye real numbers. When pos-sible use a calculator to verifu your answer. (Exumple 9)
77. f8 +f -{izxT + 2\/t - 2\/5
7s. \48 - \,6 + fi - \4t zt,5 - zfl
ts. (-zx6)(s:,G) 30ft ao. (-:rD)(-zx6)axG
Evaluate each radical expression. Use a calculator to check.(Exumple 5)
:s. l6oo:tr za. fqoo2.o 37. V-8 2
3s. V4 2 40. gA2
qz. r f Jt tq +r. r , /o.oro.rY I t )
. I 84s. .'/ - -
v , 1000- I / 5
ts. l/ss zz
se. VA q
l44r . \ l ;213
u. .v6zso.s
n. t14d s
52. a (b+r ,+ l ) - t /2
\,/il + |55. {F'3/s
sz, J- r tt.-v x
54. _ 4\,G -4x3t')
*il*:
sl. (3v6X+{sa) eo"
* r ( - s r A \ z r s" * ' \ ' ' " ) ' '^ ^ a
85 ' V l8d + 1 /2o+: : - "0'
- \ \ / Y
8 7 . 5 + Y x :
n. (-zf e)(zrG) -ze
t+. (zfi)'z+s
s6. f2k1 ={3r ' * 'V i- . , t f i
88. a + Vb;
s2. \/G . \fr V2ooo
s4 . \A .Y4Vn' ge. f/zo. \6{5r;,
sB. "V l/2a*A
Write each expression involving rational exponents in radical nota-tion and each expression involving radicals in exponential notation.lExumple 6)
49. rc213 {Kf 50. _2314 *F f l . 3y-3ls L
Bg. \50f + f q5x3 s*\,rtu uso. Vt6oo + t'/sql soV2u
Write each expression using a single radical sign. Assume that allvariables represent positive real numbers. Simphfy the radicandwhere possible. (Example Ir,1
sr. tA . ft vn%. f ,6. \%X,Et;
ss. fi. \/2*y {+r','
gl. 'V t/l tfi
Solve each problem.
s6. \/7;7 (x3 +.yr1r/:
Simplfy each radical expression. Assume that all variables repre-sent positive real numbers. (Erample 7)
sl. f ta"t q'
SS. "W z),3
".ffifu, fff-Write each radical expression in simplified form. Assume that allvariables represent positive real numbers. (Example 8)
6s. fn2rt 66. xTs$ 6i.++ 4u. L=*Y s r V 7
at f * , ,un. V; Y' ,o. lX ',Ft'. t6zt s 72. tAZ.rr z
B. V-2sor 74. V-uo, ,t.#+ 76.5;r \ 2r - 2u V3i
I^!Jx
\25
\4t5
s8. f n tya t r r '
60. \/rzsr8 s,"
".^l*+"6a, i@' :Y y ' . v -
99. Economic Order Quantity Purchasing managers use theformula
_ l2AS- V I
to determine the most economic order quantity E for partsused in production. I is the quantity that the plant will use inone year, S is the cost of setup for making the part, and 1 isthe cost of holding one unit in stock for one year. Find E ifS : $ 6 0 0 0 , A : 2 5 , a n d I : $ 1 4 0 . 4 6
100. Piano Tuning The note middle C on a piano is tuned so thatthe string vibrates at 262 cy cles per second, or 262 Hz(Hertz). The C note that is one octave higher is tuned to524H2. Tuning for the 11 notes in between using the methodof equal temperament rs 262 . 2'/t2, wheren takes the valuesI through 1 1. Find the tuning rounded to the nearest wholeHertz for those 11 notes.27 8, 294, 3 t2, 330, 3s0, 37 l , 393, 41 6, 441, 467, 495 Hz
l0l. Sail Area-Displacement Ratio The sail area-displacementratio S is given by
^ 16Ar :
;tF.
where I is the sai l area 1ft2.; and d is the displacement ( lbs).S measures the amount of power available to drive a sail-boat (Ted Brewer Yacht Design, wwwtedbrewer.com). Ra-t ios typical ly range from 15 to 25, with a high rat io indicat-ing a powerful boat. Find S for the USS Constitution, whichhas a displacement of 2200 tons, a sai l area of 42,700 ft2,and 44 guns. 25.4
102. A Less Powerful Boat Find S (from the previous exercise)for the Ted Hood 5 L It has a sail area of 1302 ft2, a dis-placement of 49,400 pounds, a length of 5 I feet, and noguns. I _5.5
703, Depreciation Rate lfthe cost ofan item is Cand after nyears its value is,S, then the annual depreciation rate r isgiven by r : 1 - (SlC)tt ' . After a useful l i fe of r years acomputer with an original cost of $5000 has a salvage valueof $200.a. Use the accompanying graph to estimate r if n : 5 and if
n : 10.50%,. 30,2,
b . U s e t h e f o r m u l a t o f i n d r i f n : 5 a n d i f n : 1 0 .47 .5"1'.27 .5"/, ,
2 4 6 8 1 0Useful lif'e (years)
r Figure for Exercise 103
BMII/ Depreciation A new BMW Z3 convertible sells for$30,193 while a f ive-year-old model sel ls for $17,095(Edmund's, www.edmunds.com). Use the formula from theprevious exercise to find the annual depreciation rate. l0.l{,2,
Longest Screwdriver A toolbox has length I, width Il, andheight H. The length D of the longest screwdriver that will fitinside the box is given by
D : Q 2 + W 2 + H \ t 1 2 .
Find the length of the longest screwdriver that will fit in a4 in. bv 6 in. bv 12 in. box. 14 in.
P.3 r i lm Exercises 37
106. Changing Raditts The radius of a sphere r is given in ternrsof its volume Vby the formula
( o'tsvl 'r '' - \
r r )
By how many inches has the radius ofa spherical balloon in-creased when the amount ofair in the balloon is increasedfrom 4.2 ft3 to 4.3 ft3? o.og in.
107. Her"oni; Formttla lf the lengths of the sides of a triangle area,b,and c, and.s : (a + b + c)12, then the areal is givenby the formula
/_@
Find the area of a triangle whose sides are 6 ft, 7 ft, and ll ft(see f igure for Exercises 107 and 108). 19.0 ftr
108, Area o/ un Equiluteral Triangle Use Heron's formulafrom the previous exercise to find a formula for the areaof an equilateral tr iangle with sides of length a andsimpli fy i t . , ,V:
4
n Figure for Exercises
For Writing/Discussion
109, Roots or Powers Which onc of the following expressions isnot equivalent to the others? Explain in writing how you ar-rived at your decision. b
".({,)^ b. VF ". t/F
d. ta/s e. (t t /s1t
I10. Which one of the fol lowing expressions is not equivalent to
f ottft Write your reasoning in a paragraph. ca . l a l . t 2 b . lab 'z l c . ab2
d. (a2b\t l2 e. *fu2
7ll. The Lost Rule? Is it true that the square root ofa sum isequal to the sum of the square roots? Explain. Give examples.No
ll2. Tbchnical i t ies l f mandnarereal numbers andm2: n,then rr is a square root of r, but if nr : n then m is the cuberoot of r. How do we know when to use "a" or "the"?
,/ \,
/ \
fOZ ana fOe
0.8
* t r .b
'jf
'6 0.4I
, i , Q
i l i O 6 .2
104.
I 05.
