ࡱ> RTQ }bjbjww 4-%%%8]\%vE"+++nupupupupupupu$x7{u+++++u4v}!}!}!+^nu}!+nu}!}!.l>sϩG%o@Zuv0vVo{'z{>s>s{Rs++}!+++++uu+++v++++{+++++++++ :  Unit 1 Laws of Exponents An exponent is a numeric or algebraic expression that indicates how many times a quantity is to be multiplied by itself. In an algebraic or numerical expression, the base is the number or variable with an exponent. The base can be a natural number N an integer Z, a fraction or a decimal Q. Laws of signs in multiplication: + x + = + " x " = + + x " = " " x + = " Any power with a positive base is positive. Any power with a negative base is positive if the exponent is an even number. Any power with a negative base is negative if the exponent is an odd number. Laws of Exponents : When two or more powers with like bases are multiplied, the product has the same base and the exponents are added. a m a n = a m + n When two powers with like bases are divided, the quotient has the same base and the exponents are subtracted. a m a n = a m " n , a `" 0 When the base has a negative exponent, the base is inverted and the expression becomes positive.  EMBED Equation.3  = EMBED Equation.3  , where m  Q + Any expression other than 0 that has the exponent 0 is equal to 1. a 0 = 1 To raise a power m with base a to a power n, multiply the exponents. (am) n = a m n To raise a product in exponential form to a power n , multiply the exponents of each of the factors by n. (a b c) m = a m b m c m where a , b , and c represent exponential expressions. To raise a quotient of two expressions to a power n , multiply the exponents of each expression by n.  EMBED Equation.3  or (a b) m = a m b m , where b `" 0 Rules for Applying the Laws of Exponents: The power of a positive base is positive. The power of a negative base is positive if the exponent is an even number. The power of a negative base is negative if the exponent is an odd number. When the base has a negative exponent, invert the base and change the sign of the exponent to obtain an equivalent expression in which the exponent is positive. When an expression preceded by a negative sign has an exponent, each term must be raised to this power, including the digit "1. Scientific Notation When we multiply a decimal number by a power of 10 (i.e. 10, 100, 1000, 10 000, etc) the decimal point is moved to the right by as many places as there are zeros in the power of 10. When we divide a decimal number by a power of 10 the decimal point is moved to the left by as many places as there are zeros in the power of 10. In scientific notation, a number is always expressed in the form a 10 n, where n is an integer and a is a decimal, such that 1 d" a < 10. The number a 10 n can be converted to a decimal by moving the decimal point in a to the right by n places , where n is a natural number and 1 d" a < 10 The number a 10 " n can be converted to a decimal by moving the decimal point in a to the left by n places , where n is a natural number and 1 d" a < 10 Procedure to express a number in scientific notation : Place the decimal to the right of the first non-zero digit. Count the number of places the decimal point was moved. Write this number as the base 10 exponent : a) this exponent is positive if the decimal point was moved to the left. b) this exponent is negative if the decimal point was moved to the left. Note : To assign an exponent to several variables or numbers , place these between parentheses. Otherwise , the exponent will apply only to the closest variable or number. Unit 2 Simplifying Algebraic or Numerical Expressions Written in Exponential Form Law of priority of operations: BEDMAS Perform operations between parentheses inner brackets to outer bracketsfirst. Perform the Laws of Exponents on any exponential terms with the same base. Next perform multiplication and / or division operations. Lastly perform addition and subtraction of like terms. Add similar monomials (like terms). Note: Two monomials are similar if they are composed of the same variables with the same exponents The number 1 raised to any exponent is equal to 1. The power of a negative base is positive if the exponent is even numbered. To simplify an algebraic or a numerical expression whose terms are in exponential form: Observe the rule of priority of operations. Apply the appropriate laws of exponents. Assign the appropriate sign to each power depending on whether the exponent is even " or odd " numbered. If necessary, convert all negative exponents to positive exponents. Unit 3 Converting an Expression, Containing a Radical, to Exponential Form and Vice Versa Radicals:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 = the radicand; " = the radical sign; n " n = the root index  EMBED Equation.3   EMBED Equation.3  Place under the radical sign only those radicands that appear under radicals with the same root index. Conjugates: Binomials containing radicals consisting of identical terms , but linked together by an opposite sign are called conjugates and their product is always a rational number.  EMBED Equation.3  A numerical coefficient is a number that multiplies a variable or an arithmetic expression. The nth root of a number x is a quantity whose nth power is equal to x. The nth power of a number is the product of n factors equal to that number. The radical sign ( " ) represents the operation which consists of extracting the root of a number. The radicand is the expression under the radical sign. When the root index = 2 , it is understood. When the exponent of a number = 1 , it is understood. If a = b and b = c , then a = c. (a n ) m = a n m To convert to exponential form a radical that is the power of a single base : Remove the radicand from the radical and use it as the base of the exponential expression. Divide the exponent of the radicand by the root index and make the resulting fraction the exponent of this base. Convert to the smallest base if the base is a number and , if necessary , simplify the resulting expression by applying the fifth law of exponents. To convert an exponential expression to a radical : Place the base of the exponential expression and the numerator of the exponent under a radical sign. Make the denominator of the exponent the root index. Calculate the radicand , if necessary.  EMBED Equation.3   EMBED Equation.3  To convert a numerical or an algebraic expression of the form a m n " (a p) to an exponential expression in simplest form : Convert the term under the radical sign to exponential form. Convert the bases to the smallest possible , if necessary. Apply the appropriate laws of exponents. Unit 4 The Sum , Difference , Product and Quotient of Numerical Expressions Containing Square Roots To break a number down into factors is to find all the numbers whose product is equal to this number. Two or more radicals are similar if they have identical root indexes and radicands , regardless of the value of the numerical coefficient. The expression 3 "2 means 3 "2 . If you need to multiply two expressions containing radicals , the commutative and associative properties allow you to group the terms of the multiplication without changing the value of the result. To calculate a product involving radicals: Multiply the coefficients of the radicals. Place the radicands with the same root index under the same radical sign and keep the multiplication sign between them. Simplify the number under the radical and multiply the extracted quantity by the numerical coefficient.  EMBED Equation.3   EMBED Equation.3  Note : When multiplying radicals , place under the same radical sign only those radicands that appear under radicals with the same root index. Note that in mathematics, (  # $ > ? @ A B b c j k  > @ z  4 p t u w y z  ާ|t|t|hEhh+H* hEhh+h?U5CJaJh?Uh?UCJaJh~hAh+h+5h+h+5>*h+ hDhLhtThD hDhD hD5 hEh56 h456h4hLhL5hL hLhL hL5 h"25-A B @ B 6 u  h & Fgd2gd4gd?U & Fgd~gd+9^9gd+ & FgdL      ,j@xg[h~h~5CJaJ jh~h~5CJUaJh:65CJaJh+h:65CJaJ$jhEhhEh5CJEHUaJ&j!N h~hEh5CJUVaJh~hEh5CJaJ jh~hEh5CJUaJh:6 h4h4h?Uh~hAh+h$ hEhh+hEhh+H*@BDFH`bdfhj#$.01JKLNOPQSTYȼ}y}}uyley]e]ehEhh~H* hEhh~h0$|5CJaJh0$|h*hlh~5h~ hlh~h?U5CJaJh?Uh:6h$hEhh:6H*hEhh:66 hEhh:6h+h:65CJaJ jh~h~5CJUaJ$j7h~h~5CJEHUaJ&jb/K h~h~5CJUVaJ$hjKLbc$% hgd~T`Tgd?UT`Tgd~ & Fgd~gd0$|T^Tgd* & Fgd0$|gd%gd?U & Fgd2gd4Y`b8W $b~ytyo hA5 h"25 h?U5h+h~5CJaJh*5CJaJ$jahEhh*5CJEHUaJ j#N h*5CJUVaJhEh5CJaJjhEh5CJUaJh~5CJaJh0$|5CJaJh0$|h?Uh*h~ hEhh~hEhh~H*+bfh.TX^ &(0Jp~ 68R| 濱✕hh$5 h$5 hh$h$ hH*h}+h%CJaJh:6h%5CJH*aJh:6h%5CJaJhh%5 hhhh5hh%h:6h%5hAh?U h?U53hT,.VXTV'pT^Tgd$ & F gd$gd$gd% & Fgd?Ufgh ) 0 { !!!!!!!!!!'!v!w!!!!!Z"""$0$$$$ %X%h%p%r%%ϼ沫ϟᲛ᲍~jh0$|5U h0$|5h2h0$|5>*hhkFh"2hh5 hh hkF5 h"25 h h hAmh+{"ha#h h ha#5 h 5 h5 h5h+h\W hlh\W h\W 5/gh0 { !!!!w!!!!""F## & FgdkFgdkF & FgdkFgd  & Fgd `^``gdgdgd:6gd\W #$$$V%X%%%&&&&''(((())c)) & Fgdf `^``gdAmp^pgd0$|gd0$| `^``gdkFgd\W  & FgdkFT`TgdkF%%%%%%%%%%%%%%&&&&&6&8&:&@&`&b&Ŀēul]JuC·?Ch h_?h0$|$j% h_?h_?5CJEHUaJjk/K h_?CJUVaJh_?5CJaJjh_?5CJUaJh0$|h2h0$|5CJaJh2h0$|5CJH*aJj h0$|h_?5EHUjTk/K h_?CJUVaJ h_?5jh_?5Uh0$|5CJaJjh0$|5Ujh0$|h_?5EHUjj/K h_?CJUVaJb&d&f&h&j&l&&&&&&&&&&&&&&&&''paMpIAIh,1"h0$|5h0$|'jh_?h-5CJEHH*UaJj`m/K h-CJUVaJjh_?5CJH*UaJh_?5CJH*aJ'j+h_?h_?5CJEHH*UaJjm/K h_?CJUVaJh_?h_?5CJH*aJ#jh_?h_?5CJH*UaJ h_?h0$|h2h0$|5CJaJh2h0$|5CJH*aJh_?h0$|H*h0$|5CJaJ''''d(n((((((((((()))))#)I)K)g)h)j)k)p)q))***+ +,,,,, ,%,&,խzvvqvqvq hrOH*hrO hf5hfhfH*hfhf5H*hfhf5hU3hfhAmhAmhAm5 h@5jhfh@5EHUjl*J h@CJUVaJ hf5jhf5UhAih0$|5h0$| h th0$| h0$|5h th0$|5+)*+ +x++,&,',u,,A--- .o.... / /gdSz^gd.gd#I & FgdrO `^``gdrO & FgdrO `^``gdkFgdrO & Fgdf&,',t,--- ...............//// / /O/Q/U/00 0:0v0x0111D22˾޺ڦޕ捃{wrmih1^ hU35 h ]5hp([hp([hp([5hp([h]5H*hp([h]5hNj,hSzhSzEHUj8J hSzCJUVaJ hNH*hpjhSzhSzEHUj7J hSzCJUVaJhSzjhSzUh#I hrOhrOhrO hrO5 h}5( /x00h111C2D2226373556B66"7#7U7V7$a$gdyB7 ^`gd;M & Fgd;Mgd;MgdM `^``gdkFgdp & Fgdp([gd]222222225555666#7$7778797:7=7>7Q7R7S7T7V7\7]7777778^ٴ˟ː~vokgkeUheh(4 hyB7hyB7hyB7hyB75 hyB75jh;Mh(4EHUjPJ h(4CJUVaJjh;MUjhyB7hyB7EHUjFJ hyB7CJUVaJjhyB7Uh;M hQK5 h;M5hyB7hV3ZhMhMhM5 hMhM h1^5 h1^hU3$V777__````hajavcxcLeNeffRg(hxhhh & F gdAigdk:Ngdr `^``gdkFgd ] & F gd(4gd(4gdM 0^`0gdyB7result must never have an irrational denominator. The denominator of the resulting expression must be rationalized as follows: Multiply the numerator and the denominator of the expression to be rationalized by the radical in the denominator of this expression. Extract the root of the denominator and multiply it by the numerical coefficient of the radical, as applicable. Unit 5 Operations on Polynomials containing Square Roots The distributive property applies to multiplication and to addition or subtraction. This makes it possible to multiply the number or variable in front of the parentheses by each of the terms inside the parentheses, without changing the value of the expression. A polynomial is a form of algebraic expression composed of several terms linked by addition and / or subtraction signs. More particularly, a binomial is a polynomial with two terms , and a trinomial is a polynomial with three terms. Binomials containing radicals consisting of identical terms , but linked together by an opposite sign are called conjugates and their product is always a rational number. To calculate the product of polynomials containing radicals with the same root index: Apply the distributive property of multiplication over addition or subtraction as many times as necessary. Perform the multiplications one by one. Reduce the radicals , if necessary. Add or subtract similar terms. To rationalize a denominator consisting of a binomial containing one or two radicals: Multiply the numerator and the denominator of the expression to be rationalized by the conjugate of its denominator. Find the products in the denominator and , if applicable , in the numerator. Simplify, if necessary, Simplify the radicals in the numerator and find the common factor, if necessary. Simplify and make the denominator positive, if necessary. To rationalize a denominator composed of a monomial containing a radical, when the numerator consists of a binomial containing a radical: Multiply the numerator and the denominator by the radical in the denominator. Find the products. Simplify the radicals, if necessary, Factor out the common factor in the numerator and simplify, if necessary. Exponents and Radicals Essential Formulae Laws of Exponents : a m a n = a m + n a m a n = a m " n , a `" 0 a " m =  EMBED Equation.3  , where m  Q + a 0 = 1 (a m) n = a m n (a b c) m = a m b m c m , where a , b , and c represent exponential expressions.  EMBED Equation.3 =  EMBED Equation.3  or (a b) m = a m b m , where b `" 0 The result of simplifying an exponential expression should not have a negative exponent. To assign an exponent to several variables or numbers , place these between parentheses. Otherwise , the exponent will apply only to the closest variable or number. Remember the Law of priority of operations and to add similar monomials (like terms). Scientific Notation : A number is always expressed in the form a 10 n , where n is an integer and a is a decimal , such that 1 d" a < 10. The number a 10 n can be converted to a decimal by moving the decimal point in a to the right by n places , where n is a natural number and 1 d" a < 10 The number a 10 " n can be converted to a decimal by moving the decimal point in a to the left by n places , where n is a natural number and 1 d" a < 10 Radicals :  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 = the radicand ; " = the radical sign ; n " n = the root index  EMBED Equation.3   EMBED Equation.3  The result of simplifying an radical expression should not have a radical in the denominator . Conjugates : Binomials containing radicals consisting of identical terms , but linked together by an opposite sign are called conjugates and their product is always a rational number.     Exponents and Radicals MTH-4105 William Low PAGE  PAGE 1 C:/My Documents/My Work/Adult Ed/Math 436 Notes/Exponents and Radicals Summary ^`^^^^^^__````aajaccdddJeLe0fDfffPghiiiTltlxlmmm,o.o0o\o`oooooooooooƾƺư h7H* h75 h75>*h"2h7 hQK5 hp5hphdvr hB5 hAi5hE>hAihk:N5hk:NhAihQKhr hm5 hU35 h"25he h(45 he5h(4h%5~h(454hiij0k`klvlxlm(nNnn,o0ooooooop ppp & Fgd79^9gd7gd7 & F gdpgdp & F gddvrgdAioooooo ppp$p*p0p2pVpXpZp\p`pbpdpxpzp|p~pppppppppppppppppplqnqqqqqqqqq jj(?K h75CJUVaJjQ#h75CJEHUaJ ji(?K h75CJUVaJjh75CJUaJ h76jC!h7EHUjh(?K h7UVjh7Uh75CJaJ h7H*h72pppppjqlq r"rrr"t$tttuu>w@wxByDyyyPzRz {{gd7 & Fgd7qqqqqqqqqrrttTubuduhuxuzuuuuu v"vfwlwxxxxxxxxxxxxxxyy:yձ˧˚٧ˁuj)*h75EHUjl(?K h75UVj'h75EHUjk(?K h75UVjh75U h75>*h7CJaJh75CJH*aJ h75 h7H*h7h75CJaJjh75CJUaJj%h75CJEHUaJ+:yy@yByDyFyjylynypytyyyyyyyyyyyyyzzzz"z$zHzԱԜԂp_!j1h75CJEHH*UaJ#jn(?K h75CJH*UVaJjh75CJH*UaJh75CJH*aJ h7H*j%/h75CJEHUaJ jm(?K h75CJUVaJjh75CJUaJh7h75CJaJjh75Uj,h75EHUjp(?K h75UVHzJzLzNzPz{{*{ | |~|||||||||||||||}}} } }}}}} }}}}¾ysysysyhys`h?UCJaJhsw0JmHnHu h?U0Jjh?U0JUh?UhLh?UCJaJ h"25h"2h"25hswjhswU h"2h\* h75 h7h7h7h75CJH*aJjh75CJH*UaJ!j3h75CJEHH*UaJ#jo(?K h75CJH*UVaJ%{|||||||||||}} } }}}}}h]hgdL &`#$gdbgd"2gd"2}}} h"2h\*hsw21h:p?U/ =!S"S#S$n% 7Dd @b  c $A? ?3"`?2"%]`I v]D `!U"%]`I v #xP=OA};UCqhwX[M\@rڊކ`BIcW츜{oތB(} 6xWhu#\NQ{Bphs{`lmʻ)Udx1)АKcH ~^fe_'=v^{٬]2cVx-06%=s_=;/tvI>FGp_]UXNΓZp3%w":h0h3oxYwCI?~Y*Dd lb  c $A? ?3"`?2t /~",~Q-P{ `!H /~",~Q-hxcdd``f 2 ĜL0##0KQ* W􀹁IRcgbR PtfĒʂT/&`b]F"L L`0ro&.dabM-VK-WMcX|"+|-J 1lS&;Jy 3 ps'2G1LH0DwBex la%4 6b;/LLJ% A2u(2t5,Āf~;c+Dd b  c $A? ?3"`?2ržGJ凧?:C 6_ `!ržGJ凧?:C 6_ (+xRKPwIZZZ E BUZٺ?CWNIk }%' y^0=)g(Ha(U-,0 {ҳ(iű`iW3ST۸Df*kr^{BZd?E5 zq3{iP0Z}Z=2]h>-ZW(C+O\ xu.A2wn"'ҝ/Kq;7jPmq)™{oY{M]Qj1yjAWi #X@]-^{Ѧ.;Ä/>r,ĶJa[=(fyO{>'$[,f< TݵC|!d)/8tmS`x"Dd $b  c $A? ?3"`?2 &^ `! &^@HD yxڝR=KA0rD,.A¤I#F8Q!V+ X VvBȯP~a62f{{0`"eHSQ#($Zy]$,qTs !G<mWhNv*;ʐORb @c112BYL%bpu?@ABCDEFGHJKLMNOPSVXWYZ[]\^_`bacdegfhjiklmonprqstuwvxzy{|}~Root Entrys FGU$Data I6WordDocumentr4ObjectPooluJ2GG_1308893600mF2G2GOle CompObjfObjInfo  %(),/03678;>ADGJMPSVY\_behknqtuvy~ FMicrosoft Equation 3.0 DS Equation Equation.39q."! a "m FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native :_1261396671@ F2G2GOle CompObj fObjInfo Equation Native  C_1308893954F2G2GOle  n' $} 1a m FMicrosoft Equation 3.0 DS Equation Equation.39q.gb3w2 ab(CompObj fObjInfoEquation Native _1261398733 F2G2G) m =a m b m FMicrosoft Equation 3.0 DS Equation Equation.39qnJ ;, a mn = a mnOle CompObjfObjInfoEquation Native f_1261398868F2G2GOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qnJ`Ā  a mn =a mn FMicrosoft Equation 3.0 DS EqEquation Native f_1261398942;"F2G2GOle CompObj fuation Equation.39qn a m FMicrosoft Equation 3.0 DS Equation Equation.39qno(*   a n   ObjInfo!!Equation Native "6_1261399443'E$F2G2GOle #CompObj#%$fObjInfo&&Equation Native '_1261399392)F2G2G   b n   = ab n FMicrosoft Equation 3.0 DS Equation Equation.39qn~x 2  a n      b n   = aOle *CompObj(*+fObjInfo+-Equation Native .b() n FMicrosoft Equation 3.0 DS Equation Equation.39q O/ numerical  coefficient   root  index_1255156332.F2G2GOle 1CompObj-/2fObjInfo04Equation Native 5_1255159716,63F2G2GOle 9CompObj24:f  radicand  FMicrosoft Equation 3.0 DS Equation Equation.39qJn a )mn = a mnObjInfo5<Equation Native =f_12551598378F2G2GOle ?CompObj79@fObjInfo:BEquation Native Cf_12551636081=F2G2G FMicrosoft Equation 3.0 DS Equation Equation.39qJF  a mn =a )mn FMicrosoft Equation 3.0 DS Equation Equation.39qOle ECompObj<>FfObjInfo?HEquation Native ImQ n  a n  b n = ab n FMicrosoft Equation 3.0 DS Equation Equation.39qQ  a n  _1255165957BF2G2GOle KCompObjACLfObjInfoDNEquation Native Om_1262430312GF2G2GOle QCompObjFHRfb n = ab n FMicrosoft Equation 3.0 DS Equation Equation.39qn' $} 1a mObjInfoITEquation Native UC_1262430313cLF2G2GOle WCompObjKMXfObjInfoNZEquation Native [f_1262430314QF2G2G FMicrosoft Equation 3.0 DS Equation Equation.39qJԁ a  b  () m FMicrosoft Equation 3.0 DS Equation Equation.39qOle ]CompObjPR^fObjInfoS`Equation Native aQ5}, a m b m FMicrosoft Equation 3.0 DS Equation Equation.39qnJ ;, a mn_1262430315OYVF2G2GOle cCompObjUWdfObjInfoXfEquation Native gf_1262430316[F2G2GOle iCompObjZ\jf = a mn FMicrosoft Equation 3.0 DS Equation Equation.39qnJ`Ā  a mn =a mnObjInfo]lEquation Native mf_1262430320`F2G2GOle oCompObj_apfObjInfobrEquation Native s_1262430317TeF2G2G FMicrosoft Equation 3.0 DS Equation Equation.39q O/ numerical  coefficient   root  index  radicand Ole wCompObjdfxfObjInfogzEquation Native {6 FMicrosoft Equation 3.0 DS Equation Equation.39qn a m FMicrosoft Equation 3.0 DS Equation Equation.39q_1262430318jF2G2GOle |CompObjik}fObjInfolEquation Native _1262430319h^oF2G2GOle CompObjnpfzno(*   a n      b n   = ab n FMicrosoft Equation 3.0 DS Equation Equation.39qn~x 2  a n   ObjInfoqEquation Native 1Table;9|SummaryInformation(t@   b n   = ab() n՜.+,0 hp  RiskLow-Mathematics S- Exponents and Radicals Title14Microsoft Office Word@bވ'@𚻆 5Z߫RVT?e) +_VR])Q 4_JI-zTtg^ wT,mիj;p|oXxs'>(vn; ΃~8wa _qY{kp}7YmC~[pn90\F9k"XsAuip> _sDd Tb   c $A ? ?3"`?2TԧwYS3(: `!TԧwYS3( H(@CxڝRK@~wM M[:D ؎.M5$Fb'I I.R) .Jqh|#B .^xݗ}޻#Pv @ FF"ar@$եi,x 21h!xOIMM޷6!'l03KBhKZ)ѮwŜ"ĸ< 3}VelX9 'XzWCywKAsʦY[KH1|3-Y]-7Wح{XMoW'թkcۭYfRrL<t৕Ύ˨k?g NS hYtUzԽ,!Q ⽾|)F`9-ڻ^n.k퍏Yr)8bRp_[|O(|wT1D|c?cFޜ S8睄N`I0崠tWY|!Z3<J"\b 0{<&rqDd tb   c $A ? ?3"`? 2}-J&緍ЈNp `!}-J&緍ЈN  yxڝR=KAHr "`XXZ8D3B":S)VZA4B4`'(Alen!`4y④FHqKOyeJ"A,|QNj `k7c`IC,G qCi01.jn[NYʨj hgǴIo“l"+|)m~m #8uxNSXunгEQ,:?>os?S5FP$_COۭnlyJSOѓiA\l a;3`NuDd hb   c $A ? ?3"`? 2Z{YuU `!Z{YuUĢ @ |xڝSK@~.mڴ`[8D EJGuh BRVmI!c]:8Up1}^.QzB,Deː+EL{ϋ!s>.nx7%@7ABvc5:fD4P 'TDd hb   c $A ? ?3"`? 2KLq, `!KLq,@ @p|xڝ=KAgv/-B8-GB|6 D"DL "[ `%JBs?nr` <B Q"GH<:\LIbIe HNjfzȢ;b k[I $?k( "';Zg2c8c|犐ASϥoDU ӹۦ ?o*~ bMLşܷ8ܙ8=]>j^[J{71wX>D!w6]/΂~Tϫ UWp:e]r1eޒ~\HG9^?1Hu)wh#x8jQl6p/Efuy K_uUe\oѬe'}[ASܺFej<8ؚ.4#}ck~5AoY4z6.BU6[Q m5bNX~onz3DdF  3 A?"?2uDH][/t>Q% `!IDH][/t> fdxcdd``~$d@9`,&FF(`TIIRcgbR VPIfĒʂT/&`b]F"LL`0ro&.dabM-VK-WMcX|"+|-+b&#.#.uf Jʇkr^{BZd?E5 zq3{iP0Z}Z=2]h>-ZW(C+O\ xu.A2wn"'ҝ/Kq;7jPmq)™{oY{M]Qj1yjAWi #X@]-^{Ѧ.;Ä/>r,ĶJa[=(fyO{>'$[,f< TݵC|!d)/8tmS`x"qDd$F  3 A?"?2 J_Gm* `! J_G@(D I yxڝR=KA0rD,.A¤I#F8Q!V+ X VvBȯP~a62f{{0`"eHSQ#($Zy]$,qTs !G<mWhNv*;ʐORb @c112BYL%bpu(vn; ΃~8wa _qY{kp}7YmC~[pn90\F9k"XsAuip> _sDocumentSummaryInformation8DCompObjy      !"#$%&'()*+,-./0123456789:<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyOh+'0@  < H T `lt|Exponents and Radicals William LowNormal William Low14Microsoft Office Word@bވ'@R@(R@z)&G>VT$m  i&" WMFC" lVT$m EMF\KhC   % % Rp@Times New Roman ͕ ѕ Ε Qdѕ Ε ϕ Е TQdѕ Ε cvfΕ ѕ lvfXG*Ax Times ew RomanGDΕ 2fΕ Ε {fΕ ldv% % %  Rp@Times New Roman 4υ ҅ Ѕ Qd҅ υ х |҅ TQd҅ υ cvfυ ҅ ,9vfXG*Ax Times ew Roman@E@LP r% % % Tx?d@E@?L\+ x + =w8388TTej@E@eLP uTTk@E@kLP tTT@E@LP+r8TT @E@LP t  TT  @E@ LP"s8  T` 3 @E@ LT x 3  TT4 k @E@ LP"i8  TTl  @E@ LP  T`  @E@ LT= +t89TT  @E@ LP a  Td  @E@ LT+ x 83  TT  @E@ LP"t8  T` - @E@ LT = e8  TT. f @E@ LP" 9 TTg F@E@g LP G TTG~@E@GLP"c8  T|@E@GL\ x + = 388  TT@E@GLP"u9 TT@E@LP n- TTc@E@crLP e-Rp@Symbol ͕ 4ѕ Ε Qd4ѕ ,Ε ϕ ѕ TQd4ѕ ,Ε cvf,Ε 4ѕ \MvfX5SymboldNkmdא7 @E@ LP a,TT? j7 @E@? LP e,TTk 7 @E@k LP =,% % % TT ; @E@ LP 5!" Rp @Times New Roman Ε ѕ xϕ Qdѕ Ε `Е ѕ TQdѕ Ε cvfΕ ѕ ,vfXG*Ax Times ew RomanD$ϕ 2f`ϕ `ϕ {fϕ ,dv% % % Rp @Times New Roman ɕ L͕ ʕ QdL͕ Dʕ ˕ 0͕ TQdL͕ Dʕ cvfDʕ L͕ LvfXG*Ax Times ew RomanDGxʕ 2fʕ ʕ {fʕ Ldv% % % % % % % % % Rp  @Times New Roman˕ @ɕ Qd˕ ȕ (ʕ ˕ TQd˕ ȕ cvfȕ ˕ LLvfXG*Ax Times ew omanYȕ 2f(ɕ (ɕ {fPɕ LLdvdv% Rp @Times New RomanvfXgwcH|,,Ax ` D̕ ku},BL Dz,{,h~p,(̕ Lݕ wҋgwdw@dw!%,h~,h~uw%,` DΕ !Ε uTudv% ( % (  % % % TTc(  @E@c LP ,TT(  @E@ LP <,% % % TX9  @E@ LPa ,% % % TT) 3t @E@e LPm 4% % % TT49 L @E@4 LP TdM9  @E@M LT a 8,% % % TT) t @E@e LPng"% % % TT9  @E@ LP rTd9  @E@ LT= a 8,% % % Tx) st @E@e L\m + n n4&"% % % TTt9  @E@t LP .TT9  @E@ LP ,- TTc  @E@c LP o- % % % TXc  @E@cy LP2.2% % % TT  @E@y LP C% % % T  @E@y LdWhen two poi_2,2H222T   @E@ y Lwers with like bases are dividedH,!'H22,2,','-!,222,2T   @E@ y 7L, the quotient has the same base and the exponents are t2,222,22,'2,',N,2,(,,222,,3222,2',",!"  T  @E@ Ldsubtracted. '22!,,,2TT  @E@ LP iTT  @E@ LP ,TT  @E@ LP r&" WMFC j-!" % % % Rp @Times New RomanL͕ ʕ QdL͕ Dʕ ˕ 0͕ TQdL͕ Dʕ cvfDʕ L͕ LvfXG*Ax Times ew omanDGxʕ 2fʕ ʕ {fʕ Ldvdv% Rp  @Times New RomanvfXgwcH|,,Ax `D@̕ ku},BLDz,{,h~p,(̕ Lݕ wҋgwdw@dw!@%,h~,h~uw%,`DΕ !Ε uTudv% ( % (  % % % TTc v @E@c_ LP ,TT v @E@_ LP y,TX v @E@_ LPa ,% % % TT 3@ @E@1 LPmo4% % % TT4 Lv @E@4_ LP gTdM v @E@M_ LT a 7,% % % TT @ @E@1 LPns"% % % TT v @E@_ LP nTd v @E@_ LT= a 8,% % % TX @ @E@1 LPm 4% % % 1  TT @ @E@1 LP"i&  Td q@ @E@1 LT n ! % % % TTr v @E@r_ LP tTT v @E@_ LP,tTT >v @E@_ LP TT? jv @E@?_ LP ,k_  TXk v @E@k_ LPa ,  TT v @E@k_ LP`" 5  TX 1 v @E@k_ LP 03 TT2  ^ v @E@2 _ LP .- TTcx  @E@c LP e- % % % TXc \ @E@cE LP3.2% % % TT Z @E@E LP C% % % Tp \ @E@E LXWhen t_2,2T \ @E@E Lhe base has a negative exponenta2,2,',2,',3-2,2,,3222,2Tx \ @E@E 2L, the base is inverted and the expression becomes 2,2,','22,!,2,222,,32!,''222,,2N,'T .\ @E@E L`positive.)22'2,TT/ [\ @E@/E LP -!" % % % % % % Rp  @Times New Roman˕ @ɕ Qd˕ ȕ (ʕ ˕ TQd˕ ȕ cvfȕ ˕ LLvfXG*Ax Times ew omanYȕ 2f(ɕ (ɕ {fPɕ LLdvdv% Rp @Times New RomanvfXgwcH|,,Ax `D̕ ku},BLDz,{,h~p,(̕ Lݕ wҋgwdw@dw!%,h~,h~uw%,`DΕ !Ε uTudv% ( Rp  @Times New Roman,Ax `D̕ ku},BLDgwcH|,,9`D =ku},99 ew95 o̕ uw,uwiڐ5 o9`v@dw!̕ 5 oLݕ tw :uw|-u9fvfvuTudv% ( Rp @Times New Roman,9`D =ku},99 gwcH|,,@9`D =ku},989 ew@9Q ̕ uw,uwiڐQ @9`v@dw!̕ Q Lݕ tw :uw|-u@9$gv$gvuTudv% ( Rp  @Times New Roman,@9`D =ku},989 gwcH|,,9`D =ku},99 ew95 p̕ uw,uwiڐ5 p9`v@dw!̕ 5 pLݕ tw :uw|-u9fvfvuTudv% ( Rp @Times New Roman,9`D =ku},99 gwcH|,,@9`D =ku},989 ew@9Q ̕ uw,uwiڐQ @9`v@dw!̕ Q Lݕ tw :uw|-u@9$gv$gvuTudv% ( Rp  @Times New Roman,@9`D =ku},989 gwcH|,,9`D =ku},99 ew95 q̕ uw,uwiڐ5 q9`v@dw!̕ 5 qLݕ tw :uw|-u9fvfvuTudv% ( Rp @Times New Roman,9`D =ku},99 gwcH|,,@9`D =ku},989 ew@9Q ̕ uw,uwiڐQ @9`v@dw!̕ Q Lݕ tw :uw|-u@9$gv$gvuTudv% ( Rp  @Times New Roman,@9`D =ku},989 gwcH|,,9`D =ku},99 ew95 r̕ uw,uwiڐ5 r9`v@dw!̕ 5 rLݕ tw :uw|-u9fvfvuTudv% ( % ( Rp @Times New Roman&" WMFC J ͕ ѕ Ε Qdѕ Ε ϕ Е TQdѕ Ε cvfΕ ѕ FvfXG*Ax Times ew RomanZDΕ 2fΕ Ε {fΕ Fdv% % %  % % % TTc @E@c LP ,TT @E@ LP ,% % % !F(GDIC k!b K!b      !% %     Rp @Times New RomanBu O }  @p XVgTv;ڱ !TT/ tw :uw|-uv; !;`v !E`vv;`v !З nu`v ٬/Zuhu >H5 > bu LPmtRpTimes New Romanu Cu  !Bu !mK} ?d u uv; ! ? @imes ew Romandv% ( TT cuu?.?ZLPaaRp Times New Romanu Cu  !BuZ !a!mK} ? u uv; !,9V=@i9x98 ew9 ? uw,dv% ( TTf 2 uu?.?oLP1rRp"Systemu Cu  ! !1!a!mK} ? u u,9,V=V=998 ew9J  uw,dv% ( % % 0 % K( ( % % " % ( " " FGDIC" % % % TTq @E@q LP s% % % TT @E@ LP,wTT @E@ LP l Rp @Times New Roman˕ @ɕ Qd˕ ȕ (ʕ ˕ TQd˕ ȕ cvfȕ ˕ LLvfXG*Ax Times ew omanYȕ 2f͕ @E@0LP % % % TX?@E@?LPor2!TTj@E@LP ET|k @E@kL\(a b) !,72!% % % TT r @E@ LPm\4% % % TT  @E@ LP Td p @E@ LT= a 8,% % % TTq r z& WMFC @E@q LPm4% % % TT  @E@ LP Td X @E@ LT b 72% % % TTY r @E@Y LPm4% % % TT  @E@ LP T|  @E@ L\, where H2,!,  TX / @E@ LPb 2  TT0 e @E@ LP`"6  TXf  @E@ LP 03 TT  @E@ LP - TTcQ@E@cLP - % % 666666666666666666666666666666666666 6 66 6  6 66 6  6 66 6  6 66 6  6 66 6 66666666666666666666  d."System??????????????????--@Times New Roman---@Times New Roman--- ,2 .+eExponents and Radicals  2 .e A2 .eMTHh  2 .e-2 .e4105 2 .e 2 .e $---2 . eWilliam Low  2 .5e ,e'2 +NeC:/My Documents/My Work/Adult Ed/Math 436 Notes/Exponents and Radicals Summary     2 e ,e',:4@Times New Roman--- 2 44:1 2 :4: ,:4''---  2 <+e 2 J+eUnit 1 2 JJe  2 JOe $%2 JseLaws of Exponentsi  2 Je  2 X+e ---2 e+eAn--- 2 e9e 2 e< eexponent ---2 enWeis a numeric or algebraic expression that indicates how many times a quantity is to be     +2 s+emultiplied by itself.  2 se  2 +e ---O2 +-eIn an algebraic or numerical expression, the  ---2 ebase--- 2 e L2 +eis the number or variable with an exponent.e  2 e 2  eThe base ca2 )en @Times New Roman------)2 +ebe a natural number  --- 2 eN --- 2 e 2 ean integer --- 2 eZ---22 e, a fraction or a decimal  --- 2 HeQ --- 2 Qe. 2 Te  2 +e ---@Times New Roman- @Times New Roman-----;2 + eLaws of signs in multiplication:  2 e ---2 e+ x + =  2 e  2 e  2 e+ 2 e '. 2 'e-...2 .e x ..: 2 :e-..A 2 Ae .2 De= +  2 Te o.2 oe+ x .. 2 e-..2 e = .. 2 e-. 2 e . 2 e-..2 e x + = .. 2 e-. 2 e  2 +e @Symbol---@"Arial------ 2 .e--- 2 3e ---L2 <+eAny power with a positive base is positive.e   2 e ,e'--- 2 .e--- 2 3e ---2 <MeAny power with a negative base is positive if the exponent is an even number.    2 e ,e'--- 2 .e--- 2 3e --->2 <"eAny power with a negative base is   2 enega42 etive if the exponent is an 2 teodda 2 e 2 enumber.   2 e ,e' 2 .e @Times New Roman------%2 .eLaws of Exponentse- @ !a.-  2 e  2 e:--- 2 e  2 +e ---2 (+e1.--- 2 (4e ---2 (<eWh +2 (Meen two or more powers   2 (e 82 (ewith like bases are multiplied G2 (L(e, the product has the same base and the  ,e'@Times New Roman------)2 6<eexponents are added. 2 6e  2 6e $ 2 6e $ 2 6e $--- 2 6'e ,e'@Times New Roman- - - @Times New Roman- - - - - - --- @Times New Roman- @Times New Roman-  - - - - 2 D+e $ 2 DOe $---2 Dsea - - - 2 >{em--- 2 De 2 De a - - - 2 >en--- 2 De 2 De= a - - - 2 >em + n --- 2 De  2 De  2 R+e ---2 _+e2.--- 2 _4e ---2 _< eWhen two po ;2 _| ewers with like bases are divided  ^2 _7e, the quotient has the same base and the exponents are  ,e'2 m< esubtracted. 2 mpe  2 mse $ 2 me ,e'- - - @Times New Roman-  @Times New Roman-  -  --- 2 {+e $ 2 {Oe $2 {sea - - - 2 u{em--- 2 {e 2 {e a - - - 2 uen--- 2 {e 2 {e= a - - - 2 uem - - - u. 2 ue-..u2 ue n .--- 2 {e  2 {e, 2 {e  2 {e ${.2 {ea ..{  2 { e?..{2 {e 0. 2 {e  2 +e ---2 +e3.--- 2 4e ---2 <eWhen t :2 _ehe base has a negative exponentV2 2e, the base is inverted and the expression becomes  2  epositive. 2 e ,e'- - - --- @Times New Roman- @Times New Roman-   @Times New Roman-  @Times New Roman-   @Times New Roman-  @Times New Roman-   @Times New Roman-  @Times New Roman-   @Times New Roman-  - @Times New Roman- - - --- 2 +e $ 2 Oe $- - - ,e,- - .  @Times New Roman- .Times New Roman- 2 mTimes New Roman- 2 uaSymbol- 2 |-"System-- - - - - '"System??????????????????-'''- - - 2 e  2 e=,e,- - .  @Times New Roman- .-Times New Roman- 2 mTimes New Roman- 2 aTimes New Roman- 2 1"System-- - - - - '-'''- - - 2 e --- 2 e, 2 e @Times New Roman--.2  ewhere m  .-- - - 2 eQ --- 2 e - - - 2 e+--- 2 e  2 +e ---2 +e4.--- 2 4e ---n2 <BeAny expression other than 0 that has the exponent 0 is equal to 1. 2 ue ,e' 2 +e - - - 2 +e $ 2 Oe $---2 sea - - - 2 {e0--- 2 e  2 e= 2 e  2 e1 2 e  2 +e ---2 +e5.--- 2 4e --- 2 <eTo raise a pow 2 eer t--- 2 em --- 2 e 2  ewith base  2 e --- 2 ea--- 2 e 2  eto a powe  2  er 2 e --- 2 en---12 e, multiply the exponents.  2 e ,e' 2 <e - - -  @Times New Roman-@Times New Roman-- - - - 2 #+e $ 2 #Oe $--- 2 #se( 2 #wea- - - 2 |em---2 #e) - - - 2 en--- 2 #e 2 #e= a - - - 2 em n --- 2 #e  2 #e  2 1+e ---2 ?+e6.--- 2 ?4e ---2 ?<deTo raise a product in exponential form to a power n , multiply the exponents of each of the factors    2 ?e ,e'2 M<eby n 2 MQe. 2 MTe --- 2 Z<e  2 ZOe $2 Zse(a b c) - - - 2 Uem--- 2 Ze 2 Ze= a - - - 2 Uem---2 Ze 2 Zeb - - - 2 Uem---2 Ze 2 Zec - - - 2 Uem--- 2 Ze  2 Ze 2 Ze $"2 Z'ewhere a , b , G2 Zh(eand c represent exponential expressions. 2 Z)e  2 h+e ---2 v+e7.--- 2 v4e ---(2 v<eTo raise a quotient 2 ve 82 veof two expressions to a power  Y2 v+4en , multiply the exponents of each expression by n.  2 v#e ,e' 2 +e - - - 2 +e $ 2 Oe $,e,- - .  @Times New Roman- .-zTimes New Roman- 2 m 2 m 2 mTimes New Roman- 2 b 2 a 2 {b 2 {aSymbol- 2 = 2  2  2  2 t 2 t 2 t"System-- - - - - '-'''- - - 2 e 2 e  2 e !---2 eor 2 e 2 e(a b) - - - 2 &em--- 2 ,e 2 /e= a - - - 2 Aem--- 2 Ge 2 Je b - - - 2 ]em--- 2 ce 2 fe, where  .2 eb .. 2 e?..2 e 0. 2 e  2 +e --eeeeddddddddddddddddcccccccccccccccccc^ 2 0@P`p2( 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p8XV~_HmH nHsH tH@`@ NormalCJ_HaJmH sH tH DA`D Default Paragraph FontRiR  Table Normal4 l4a (k (No List jj do Table Grid7:V04@4 LHeader  !4 @4 LFooter  !.)@!. L Page NumberH2H AP Balloon TextCJOJQJ^JaJPK![Content_Types].xmlj0Eжr(΢Iw},-j4 wP-t#bΙ{UTU^hd}㨫)*1P' ^W0)T9<l#$yi};~@(Hu* Dנz/0ǰ $ X3aZ,D0j~3߶b~i>3\`?/[G\!-Rk.sԻ..a濭?PK!֧6 _rels/.relsj0 }Q%v/C/}(h"O = C?hv=Ʌ%[xp{۵_Pѣ<1H0ORBdJE4b$q_6LR7`0̞O,En7Lib/SeеPK!kytheme/theme/themeManager.xml M @}w7c(EbˮCAǠҟ7՛K Y, e.|,H,lxɴIsQ}#Ր ֵ+!,^$j=GW)E+& 8PK!Ptheme/theme/theme1.xmlYOo6w toc'vuر-MniP@I}úama[إ4:lЯGRX^6؊>$ !)O^rC$y@/yH*񄴽)޵߻UDb`}"qۋJחX^)I`nEp)liV[]1M<OP6r=zgbIguSebORD۫qu gZo~ٺlAplxpT0+[}`jzAV2Fi@qv֬5\|ʜ̭NleXdsjcs7f W+Ն7`g ȘJj|h(KD- dXiJ؇(x$( :;˹! I_TS 1?E??ZBΪmU/?~xY'y5g&΋/ɋ>GMGeD3Vq%'#q$8K)fw9:ĵ x}rxwr:\TZaG*y8IjbRc|XŻǿI u3KGnD1NIBs RuK>V.EL+M2#'fi ~V vl{u8zH *:(W☕ ~JTe\O*tHGHY}KNP*ݾ˦TѼ9/#A7qZ$*c?qUnwN%Oi4 =3ڗP 1Pm \\9Mؓ2aD];Yt\[x]}Wr|]g- eW )6-rCSj id DЇAΜIqbJ#x꺃 6k#ASh&ʌt(Q%p%m&]caSl=X\P1Mh9MVdDAaVB[݈fJíP|8 քAV^f Hn- "d>znNJ ة>b&2vKyϼD:,AGm\nziÙ.uχYC6OMf3or$5NHT[XF64T,ќM0E)`#5XY`פ;%1U٥m;R>QD DcpU'&LE/pm%]8firS4d 7y\`JnίI R3U~7+׸#m qBiDi*L69mY&iHE=(K&N!V.KeLDĕ{D vEꦚdeNƟe(MN9ߜR6&3(a/DUz<{ˊYȳV)9Z[4^n5!J?Q3eBoCM m<.vpIYfZY_p[=al-Y}Nc͙ŋ4vfavl'SA8|*u{-ߟ0%M07%<ҍPK! ѐ'theme/theme/_rels/themeManager.xml.relsM 0wooӺ&݈Э5 6?$Q ,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧6 +_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!Ptheme/theme/theme1.xmlPK-! ѐ' theme/theme/_rels/themeManager.xml.relsPK] t. :E @Yb%b&'&,2^oq:yHz}}!"$'()+-?ACDEGhh#) /V7hp{} #%&*,.@BF   !#,.=QS  '''H([(](`(s(u(++++ , ,,/,1,4,G,I,,,,,,,t.:::::::::::::::::::::::AELN!!@  @H 0(  0(  B S  ?iv---------r.u.^inoyzFHOP')  r v   c n :>qxpy   rxz SWDIOZ!"o"v"##%%^'i'''+)4)W)b)**++P,Z,,,-&----------r.u.3333333333333333333333333333333333333333333333333Vkn3yFN $      ")Pqw22\b:T/=T+ *%B%&&k''''''H(^(`(v(++++,4,J,,,,,----------r.u.#$aatt&'  !!FGHIJJ %yybb\ \         5 5 !!")Pqw22````/=Tbbxx99 !!!!9#9#f$f$&&V&V&''H(^(`(v(+++,4,J,,,,,--------------r.u.=snNM 80kP fa_ oZ|zaW vS;DYwcf/%%Tz4(Xq,y( ?~ )}\1)ί#)7oj:8s=ؕq?@JMxNj5]XQ8 ZPfT!wn42ApҨEr7x}twxx;f{IEK|"$dP|#^`56CJOJQJ^Jo(hH. ^`hH. pp^p`hH. @ @ ^@ `hH. ^`hH. ^`hH. ^`hH. ^`hH. PP^P`hH.9TT^T`OJQJo(hH9TT^T`56o(hH.9pp^p`OJQJo(hH9@ @ ^@ `OJQJo(hH9^`OJQJ^Jo(hHo9^`OJQJo(hH9^`OJQJo(hH9^`OJQJ^Jo(hHo9PP^P`OJQJo(hH TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. TT^T`o(hH. TT^T`o(hH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.h TT^T`o(hH.h ^`hH.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.9TT^T`OJQJo(hH9 TT^T`o(hH.9pp^p`OJQJo(hH9@ @ ^@ `OJQJo(hH9^`OJQJ^Jo(hHo9^`OJQJo(hH9^`OJQJo(hH9^`OJQJ^Jo(hHo9PP^P`OJQJo(hH TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. TT^T`o(hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.9TT^T`OJQJo(hH9TT^T`56CJOJQJo(hH.9pp^p`OJQJo(hH9@ @ ^@ `OJQJo(hH9^`OJQJ^Jo(hHo9^`OJQJo(hH9^`OJQJo(hH9^`OJQJ^Jo(hHo9PP^P`OJQJo(hH TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.9TT^T`OJQJo(hH9 TT^T`o(hH.9pp^p`OJQJo(hH9@ @ ^@ `OJQJo(hH9^`OJQJ^Jo(hHo9^`OJQJo(hH9^`OJQJo(hH9^`OJQJ^Jo(hHo9PP^P`OJQJo(hH TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.T^`OJQJo(hHT^`OJQJ^Jo(hHoTpp^p`OJQJo(hHT@ @ ^@ `OJQJo(hHT^`OJQJ^Jo(hHoT^`OJQJo(hHT^`OJQJo(hHT^`OJQJ^Jo(hHoTPP^P`OJQJo(hH TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.QQ^Q`56CJOJQJo(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH. TT^T`o(hH.h ^`o(hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.9TT^T`OJQJo(hH9 TT^T`o(hH.9 \\^\`o(hH.9@ @ ^@ `OJQJo(hH9^`OJQJ^Jo(hHo9^`OJQJo(hH9^`OJQJo(hH9^`OJQJ^Jo(hHo9PP^P`OJQJo(hHs=!wn~ )j:8dP|aW4(oa\1cfZPfN#)7{JMP y(x%%vS;}twq?2ApEK|Er]XQM =sk 500ReT400ZHu        ~        N        Sx        ].c        4        T        (d4      Θ        ~        dόW       N        ~        >8E6b       d       4        ~        ~        ~        ].c        4        3+        4        ~        1        dBbJ      ;Ey|{f2U3L7 \W .Am~B*T?UE>,1"+{"a#:c$9q$%+}+7yB78bY8_?b@Dk#I^0JQKoK;Mk:N OrOmQtTSUV3Zp([A[ ]1^]cicdodvr|r tvE"y5z0$|%5~:~].Eh2AMAizT[fO4_L- fK?Sz{e:9AP"24qGswWe_bO{|/G:64(4(ul30o1m#6ZX$@NbrkF}p Hc?f\*p#I--@hw mm~~HHHH H H H HHH[[t.@ @ @(@<@"&P@*X@0d@4l@^UnknownG*Ax Times New Roman5Symbol3. *Cx Arial5. .[`)Tahoma;Wingdings?= *Cx Courier NewA BCambria Math"1hKf Kf&S&S!SSx4d-- 2QHP?L2!xxExponents and Radicals William Low William Low                          F'Microsoft Office Word 97-2003 Document MSWordDocWord.Document.89q