' +JJJJ ?\>m0M='+l> /+l   d]@ŵLҦ]]LF L}BBL] X  ` 鷎귭෍ᷩ췩緈JJJJxL Lȿ L`lJJJJJ IL `L巬 췌`x (`(8`I`B` ``>J>J>VU)?`8'x0|&Hh VY)'&Y)xꪽ)' `Hh`V0^*^*>&` aI꽌ɪVɭ&Y&&Y& 꽌ɪ\8`&&꽌ɪɖ'*&%&,E'зЮ꽌ɪФ`+*xS&x'8*3Ixix&& 8  '  & x)*++`FG8`0($ p,&"`K ߼ 켩)```K ߼ 켩)`ij  !"#$%&'()*+,-./0123456789:;<=>?ֽ0LL𻽌ɪLFFB 췍@귎鷎lθҸ! 绠Ul\k 续k 续kUl\/8BθҸ!LLT L X -L[`HELLO:HH9 8L80^݌Hh ü ü݌ ռ ռ ռA ļD ļ? ļAEDE?HJ>h Լ ռ ռ ռ`HJ>݌h Hh݌`5`?`ȠHIHHHHhHH݌hHhHh݌H6 VDP (ED Z $0x8x D- ܸDD# H8`?E Vk *f???0xE Hh D#-EEE8` D ܸx D - ܸx8`-0ݩ?ʥD EEE`   LDcpq` [` ~  Lֽ0LL𻽌ɪLFFB 췍@귎鷎lθҸ! 绠Ul\k 续k 续kUl\/8BθҸ!LLT L X -L[`Ӝu`".Q`pNФbptťܥm2<(-Py0\|e<6e<g< JJJJj귍hI  aUL@ kU8  L  Q^R(jQ0l^l\  wUuW ԧ H h@ [_ /Q֩b_L`L[LLL`ª`LQLY[LXLeLee ўQH\(h0L& Ꝥ$`( R \ZLl8 ўR HH\`\Z[YS6`LxQɿu3'RͲʎRʎ]]]ɍuL͟ɍ}RLRɍg^H8 ^hZLɍR LͲɊRR% QLܤͲ Z@ -^ ş\[Z QY\[Z8`l6Lş_Ȍb_Ͳ] )Y h( ֭ͲLɍ [LLĦ__ ^ 9 LҦ3 9 a   0LjLY u< (_9 ˭ɠuɠK_9 ?LˆʎõĵL õ ĵµ aµ`` L̦µ_bJLuLz`  ȟ QlXJ̥KlV  ȟ QlV eօ3L e3L &RL &QL d L4 Ne)n `@-eff ``` f`L . tQLѤ LҦL` OPu d L Ne)noon 8ɍ` ^f\õL ^NR  RΩLҦ)\Z ʽ LHv 3h`0h8` [L NС õ`A@` ŵL^Lõ`  \ 濭0 \  ȟ Q ^\lZl^?cqH şch`fhjõĵ@OAP`u@`@&`QR`E Ls  @DAE@u`8` %@ @A@`@`@A`Mµ ) LЦ`8@AWc@8@-@HAȑ@hHȑ@ȑ@hHȑ@Ȋ@ch8&ȑ@Hȑ@Ah@LHȑ@ȑ@ htphso`hMhL`9V8U897T6S67`SAV RU LOCUNLOCCLOSREA WRIT OPE MONOMOPR BLOA p!pp p p p p`" t""#x"p0p@p@@@p@!y q q p@  LANGUAGE NOT AVAILABLRANGE ERROWRITE PROTECTEEND OF DATFILE NOT FOUNVOLUME MISMATCI/O ERRODISK FULFILE LOCKESYNTAX ERRONO BUFFERS AVAILABLFILE TYPE MISMATCPROGRAM TOO LARGNOT DIRECT COMMANč$3>L[dmx- ( [ խŠ-@跻~!Wo*9~~~~ɬƬ~_ j ʪHɪH`Lc (L ܫ㵮赎 ɱ^_ J QL_Ls贩紎 DǴҵԵƴѵӵµȴ 7 ַ :ŵƴѵǴҵȴµ納贍﵎ٵ്ᵭⳍڵL^ѵ-I `  4 ò-յ!  8صٵ紭ﵝ 7L (0+BC  7L HH`LgL{0 HH` õL H hBL BH [ h`Lo õ ڬL B ڬ LʬH hB@ յյ [L (ȴ) ȴ 7L L ( L (ȴL{ƴѵ洩ƴǴҵ 7 ^* B0 HȱBh ӵԵ 8 L8 ݲ` ܫ  / / ED B / / ]ƴS0Jȴ ȴ)  紅D贅E B ƴ  / 0L Ν `HD٤DEEhiHLGh ` ŵBѵ-` ѵB-` ܫ XI볩쳢8 DH E𳈈췍Ȍ X0L JLǵBȵC`,յp` 䯩 R-յյ`յ0` K R-յյ`ɵʵӵԵ` 4 K ( ѵҵLBȱBL8` DBHBH : ַ޵BȭߵBhhӵԵ RBܵmڵ޵ȱBݵm۵ߵ` 䯩LR˵̵ֵ׵`êĪLR E( 8` R` ELRŪƪ`췌 յյI뷭鷭귭ⵍ㵍跬ª 뷰` Lf ݵܵߵ޵ ^`8ܵ i B8` 4L ֵȱB׵ ܯ䵍൭嵍 ` DȑB׵Bֵ  ַ յյ`굎뵎쵬 뵎쵌``õĵBCõĵ`µµ`L õBĵCصص Qƴ0"Bƴ 󮜳` 0۰ϬBƴ8`i#`ЗLw!0>ﵭ` m ﳐ 7i볍 8 ЉLw`H h ݲL~ `浍국䵍뵩嵠Jm赍嵊mjnnn浈ۭm浍浭m䵍䵩m嵍`"L ŵ8ŵH ~(` d ֠z#?Ϡ\\zz]]`S5::26::"++ERROR++";::(222):"AT LINE #"(218)(219)256:"IN AM5.3-3":4NN$((XX)):NN(NN$):(NN)(NN)XXXX10025000:40uXXX(99):XX(XX101.5)(101.5):XXXX2:XX(XX102.5)(102.5):X$(XX):XX(X$):XX999XX1003000042uAN0:I16:(X$,I,1)"0"(X$,I,1)""AN144u:AN30000:4(222)253ĦH@found as many decimal@D2H@places as we needed.@2D2H@Rounding is the same as@D2H@in long division.@6V23H@= @I@"B;E"."U;I$:3:XXR(202)112:(XX)((XX))15000:AR(3)1R(2):54aXXX(999):X$(XX):X$(X$,1,6):XX(X$):AR(3):(X$)625000:0:HH10U:"@11V2H@<8> Multiply "U"*"H" and@D6H@compare the result@D6H@with "W". Here@D6H@they are the same,@D6H@so we are finished.":JHU:"@17V34H@"J|36340:350:"@10V2H@If there had been a@D2H@remainder, we would have@D2H@continued until we had@D2Z16220,144220,123266,123:221,144221,123:WG100ZZ:U(W(H10)):600:"@12V6H@Divide "W" by "H"0@D6H@and write the@D6H@quotient, "U", over@D6H@"ZZ". Substitute "U"@D6H@for the ending 0 in@D6H@the trial divisor."26"@8V37H@"U"@16V30H@"U:340:3500YYF:"@16V14H@"G".@"32(G100)(G10)"H@"G:224,123245,12316340:350:"@8V2H@<7> Double "B;E" and append@D6H@a 0 to form a new@D6H@trial divisor, ":H2(B10E):"@10V21H@"H"0.@16V28H@"H"0":J1114:"@36H"I"V@"ZZ;:680:"@2B@ "::"@16V33H@ "G;ZH8V@"N"@13V29H@"N/6"@10V2H@<6> Multiply "E"*"D10E" and@D6H@compare the result@D6H@with "SS".":FE(D10E):"@14V"32(F100)"H@"F:600:"@12V17H@Since "F"@D6H@is smaller than "SS",@D6H@accept "F" and@D6H@subtract it from "SS",@D6H@leaving "?06GQ1S"@D"32(SS100)"H@"F;I$.6600:"@12V17H@Since "F"@D6H@is larger than "SS",@D6H@reduce "E" by 1 in the@D6H@quotient and divisor@D6H@and try again.@I34H8V@"N"@13V29H@"N"@I13V"32(SS100)"H@"SS"@D32H@ "/6EN:E(D10E)Q100YY14030:340:350:"@34above "YY" and@D6H@substitute it for@D6H@the ending 0 in the@D6H@trial divisor.@8V34H@9@29H5D@9".6340:350:E(D10E)Q100YY14040:NE1:FE(D10E):"@10V2H@<6> Multiply "E"*"D10E" and@D6H@compare the result@D6H@with "SS".@I"32(SS100)"H13V@"SH@Write the answer "E"@D6H@above "YY" and@D6H@substitute for the@D6H@ending 0 in the@D6H@trial divisor",6"@8V34H@"E"@29H13V@"E:14030`-6E9:"@10V6H@Since the quotient@D6H@has two digits we@D6H@know it is too large@D6H@and use 9 instead.@D6H@Write 9 680:"@2B@ @D33H@"YY"@31H@"1(SS100))SS" ":600:213,112213,99245,99:214,112214,99:"@14V2H@<4> Find the first trial@D6H@divisor by doubling"g,6"@6H@"B" and adding a 0.@28H13V@"D"0":340:350:"@9V2H@<5> Divide "SS" by "D"0.":E914020:"@10V6@<@R@"X"@D6H@below "X", and its@D6H@square root above "X".@31H8V@"B"@3DB@"A:340:HT2:VT8:BL26:LN10:350:"@9V2H@<3> Subtract "A" from "X"@D6H@and bring "YY" down to@D6H@a new line."*6SSQ100YY+6"@13V31H@"Q"@11V33H@"YY;:680:"@2B@ @D33H@"YY;:7H@"XX:600)6"@8V2H@<1> Rewrite with the@D6H@digits grouped in@D6H@pairs, working out@D6H@from the decimal@D6H@point left and right.@10V29H@"RD$R$R$R$R$R$R$R$"@D31H@"X" "YY"."ZZ"@2U35H@.":600:"@14V2H@<2> Write the largest"*6"@6H@perfect square @G76,3:(4)"RUNAM5.3-2"i(630000:ZZ((X$,5,2)):YY((X$,2,2)):X((X$,1,1)):A1:I1X2:IIXAI )6:BA:AAA:QXA:D2B:E((Q100YY)(D10)):H19:H2270:V135:V2155:500:V259:500:"@5V2H@Compute the@D2H@value of:@15H@"RD$R$R$R$R$R$R$"@D1@"(16)"@R@";d'2R$R$R$R$R$"@D6H@"X:LN14:HT26(T3)7:V16:355:Y(T$(BL))ĺ"@17V1H@"C$:13040'2FO1:"@17V1H@"W$" the correct@D1H@answer is "Y"."(236261,(36261)FO:340:HT1:VT7:BL24:LN13:350:"@4V1HI@"20)"@D1H@"20)I$:P:410:26:245O1:"@17V1H@"W$" the correct@D1H@answer is "Y".":13040 '2"@7V1H@Select the LARGEST value@D1H@SMALLER THAN the value@2D1H@of@12V1H@Use the "(91)(2)","(1)"] keys to@D1H@light the correct@D1H@choice, and "(91)"RETURN] to@D1H@select your choice.@L9V4H0:"@8V1H@What is the value@2D1H@of "RD$R$R$R$R$R$"@D@?@12V1H@Enter your answer from@D1H@the keyboard. Press ";%2(91)(2)"]@D1H@to backspace. Press@D1H@"(91)"RETURN] when finished.@6H10V@"X:HT13:VT10:MX12:L1:620:INYĺ"@17V1H@"C$:130408&2F"Y] for yes;@D7H@"(91)"N] for no."t$2K(16384):KK128:K78K8913017:K78ININK89ĺ"@15V1H@"C$:13020$2PO$"":IN1PO$"not "$2FO1:"@15V1H@"W$" Interpolation@D1H@will "PO$"be necessary."p%2340:HT1:VT7:BL24:LN13:350:IN1303(I10)::YN}#2"@I4V1H@"24)"@D1H@"24)"@5V2H@Find"RD$R$R$R$R$R$"@8HD@"X;I$:7,51178,51:178,32178,159:179,32179,159#$2"@8V1H@Can you find"RD$R$R$R$R$R$"@D15H@"X"@21H@from@D1H@the table WITHOUT the@D1H@need for interpolation?@12V1H@Press "(91)(X)):IN1:T2Y(A$(X))10:XX100:Y((Y))"2T3NR(13)10:XX(1):XNX:X(X102.5)(102.5):IN0:Y(A$(N)):U1:I1124:T$(U)A$(I):UU1:"2T4YR(14)10:XYY:IN1#2T5NR(13)10:XR((N1)(N1)NN1)NN:IN0:I114:T$(I)51)013000:12007!.320:KK128:K89K7812000:!2"@I4V26H@(SAMPLE) @D26H@ X X@G@2@R@ "RD$R$"@5V37H@X @I5V@":I1124:B$(I):!2M4((36251)0):C3:24576,0!2P195((36251)0):FO0:400:TR(4)1:P1T1."2XR(14)10:Y(A$4 O " The answer is "AN"."; P (36251)03420:" Would@D2H@you like to see a sample problem@D2H@worked for you? (Y/N) @I@ "I$;:12000:"@B@"(K):K89HT1:VT4:BL38:LN16:350:14000!\ 36260,(36260)FO:340:HT1:VT4:BL38:LN16:350:P:(362finish:"13)I$"@L13V1H@"RD$R$R$R$R$R$R$"@RD3H@"XX"@10H@=":365L HT12:VT12:MX13:L2:620:AN(XX):AN(AN10A.5)(10A.5):AN$(AN):AN(AN$):HT2:VT15:LN4:BL36:350:INANĺ"@17V2H@"C$:3420 N "@15V2H@"W$;:FO1:TM0ĺ" Try again":TM1:3402270:V135:V283:500:"@I5V2H@Compute the square root shown below "jI IT$"place. ":A1IT$"places.">J "@2H@using paper and pencil (NOT a @D2H@calculator!) and round your answer @D2H@to "A" decimal "IT$" Enter your @D2H@answer when you t is a complex procedure."10)"@D2H@Please return to the Mode Selection @D2H@menu and select the 'Discussion' or @D2H@'Example' learning style to proceed.@D2H@"36):310 P41200:14000:310EH P195((36251)0):P1P1:FO0:400:TM0:25000:H19:H"@2H@we look at the next portion of the@D2H@dividend.":310 P11200:I$:HT1:VT4:BL38:LN16:350:"@I7V2H@"36)"@D2H@Computation of square roots is much @D2H@easier to learn from examples than " "@2H@from rules taken out of context -- @D2H@iX1100:X,I:340:350J "@24H7V@4@2D16H@4@11V2H@In this process we try to identify@D2H@partial square roots which square to@D2HI@less than or equal to@I@ the portion of@D2H@the dividend being analyzed. Any@D2H@remainders are then processed when";L 2H7V@3.@I@4@I2D14H@3.@I@4"I$:340:HT2:VT11:BL36:LN8:350:"@11V2H@This leaves us with a small@2D2H@remainder, which represents@2D2H@additional decimal positions we@2D2H@could calculate for more precision."I "@24H7V@4@2D16H@4@2U25H@";:I19:".";:st digit is 2, since 2 * 2 =@D2H@4."F "@I9V13H@2@2U19H@2"I$:600:"@14V6H@The second digit is 3, since 23@D2H@* 23 = 529.@9V13H@2@I@3@I2U19H@2 @I@3"I$:600:"@15H15V@We go to the tenths@D2H@position and use 4, since 23.4 *@D2H@23.4 = 547.56."H "@2equal the first two digit@D2H@pairs of the dividend (here,@D2H@21 * 21 = 441).":310$D H19:H2270:V134:V2156:500:V251:500:129,67129,80:130,67130,80:129,67203,67:"@5V2H@Here, matters are more difficult:@9V19H@5 48.00 00":600:"@13V2H@The firsor/quotient must square to@D2H@equal the first digit(s) of the@D2H@dividend (in this case, 2 * 2 = 4).":340:HT2:VT14:LN5:BL36:350:"@10V21H@1@2D17H@1@2D2H@Now, we guess that the first two"T@ "@2H@digits of the divisor/quotient must@D2H@square to H@is to make the divisor and the@D2H@quotient the same number:"< 129,91129,104:130,91130,104:129,91161,91:600:"@10V19H@2@2D16H@2@2F@4 41@14V2H@Write the dividend as digit pairs.":600:"@15V2H@We guess that the first digit of the"> "@2H@diviot.@4DL6H@"RD$R$R$R$"@RD3B@361 = 19 ";2 "and 361 @G@/@R@ 19 = 19@2DL6H@"RD$R$R$R$"@RD3B@441 = 21 and 441 @G@/@R@ 21 = 21":310?: H19:H2270:V134:V2156:500:V274:500:"@5V2H@We can set this up as a long@D2H@division problem of sorts. Our goal@D22B@49 = 7 and"Y( "@20H13V@ 7 * 7 = 49@2DL8H@"RD$R$R$"@RD2B@81 = 9 and 9 * 9 = 81":310&0 H19:V150:V284:H2270:500:V191:V2140:500:"@7V2H@This also means that if we divide a@D2H@number by its square root the result@D2H@will also be the square roP23000% M3100,3200,3300,3400E P3110,3120,3130,3140,1200& H19:V150:V284:H2270:500:V191:V2140:500:"@7V2H@We already know that the square root@D2H@of a number multiplied by itself@D2H@gives us the original number.@13V8HL@"RD$R$R$"@DRP0:400:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D8B@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> RETURN TO CONTENTS@I12H6D@WHICH (0-4) ??"I$:MN0:MX4:300:K0İ412:26:(4)"RUNAM5.3"MK:P1:36320M,(36320M)1:400:410 A$(24),T$(14),B$(24)UI1124:A$(((I)104.5)(104.5)):I16A$"4.0000"r(A$)6A$A$"0":1003A$(I)A$:B$(I)"@26H@"(I)(9)((II))(9)A$:RD$"@UL@"(16)"@R@":R$(12):(24576)2813000L(36251)03400M0:ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX630;t622v(PS0)620:IN$"":I0PS:IN$IN$((512I))::IN(IN$):"@R@":I1400::C(24576):35339:I$"@I@":W$"WRONG.":C$"CORRECT.":R(X)((1)X)1:X(X)(1)X:N$(2)","(1):(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32::16368,0gnPSMXĺ"@"HTPSL"HI@"((512PS))I$;p328:(PSMX)(K141)(K136)624:PSMXĺ"@"HTPSL"H@"((512PS));qK141630:K136PSĹ511PS,32:PSPS12rKK128:K47K58K45K46 light the correct@D2H@choice, and the "(91)"RETURN] key to @D2H@select your choice.":H19:H2270:V1124:V2156:500XI11500::K(16384):MK155321:bTPD(4)5:TPTP(TP10)118:g2(TP10)(TP15)3(TP138)4(TP133):<l35400,Lsh each part."1 oH19:H2270:V1116:V2156:500[ "@2H1V@"C"@5H@"P(P10)"@2HD@"M:3:k 31051:414v 30976 "@21V1HLI@"19)"@RI@": H11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2:u"@16V2H@Use "(91)(2)","(1)"] keys to# iKY149VV1:BLBL1:VV2355N kKY136VV1:BLBL1:VV1VV2:BLLNW l359 m"@15V2H@Use the number keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Use "(91)(2)"] to backspace and@D2H@"(91)"RETURN] when you fini45:"@22V1HI@"36)"@I@":I ^"@R0K"VT"V@";:I1LN:"@"HT"H@"BL)""::h `VT15:HT1:BL38:LN5:350 bI2051:I:2:I$38)I$:I:2:38):: cBL1:VV1:V2V1LN1 g"@"V"V"HT"HI@"T$(BL)I$:328:KYK:16368,0:"@"V"V"HT"H@"T$(BL):KY141ı412:K1551200:26:(4)"RUNALGEBRA 5"6 H16368,0S IK(16384):K128329:w JPP(K149)(P0)(K136):332 K:331 L63900 MK136412:K149410:1300 T"@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINUE"I$:16368,0 Y320:K160327(36251)0323:K21K8311:K21PP1:410O >K8PP1:412:P1200^ ?400:1300{ @ZH(1):328:(36251)ı A(P(K136K149)(M4))305:K155ı C"@40X40YN@";:328:K155K205ı D16368,0:326 E:325 F63900( GU-24577:3000063900) 100080:1002:Z,320:KK176:KMNKMX300:d1307o2:306{36390043126"@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE"I$73202 8KK128:K              6""#"(218)(219)256:"IN AM5.3-2":S>3000:TP0:I1(IN$):(IN$,I,1)"."TPI[>:> TP$"":I1(IN$):(IN$,I,1)" "TP$TP$(IN$,I,1)> :IN$TP$:>">6>6>">">6>6">">6>R0:L0:I120:DAN(M(P))(II):D(D)LI:UDL=:U0U1L0L1TR1R==27:HT1:VT7:BL25:LN7:29:"@7V1H@What is the length in@D1H@DECIMAL FORM to 2 decimal@D1H@places?":=(222)253Ħ#>::3::"++ERROR++";::(222):"AT LINE 's third side: ":<I13Q<B(I)R(15)1:X0I1:B(I)B(X)X10::142v<::B(3)B(1)B(3)B(2)141:<3:O$"RUNALGEBRA 5"<AN(1)B(3)2B(2)2:AN(2)B(3)2B(1)2:AN(3)B(1)2B(2)2:I13:AN(I)(AN(I)104.5)(104.5):0=TV;B$(3)"??"Ē1:269,131207,59:3:207,131207,59:207,131269,131:208,131208,59;5:209,123215,123215,131:"@I4V27H@ SIDES ARE @D27H@NOT TO SCALE"I$:<7,51185,51:185,32185,159:186,32186,159:"@4V1H@Find the length of the @D1H@"H$"159:7,51179,51:"@5V1H@Find @LU@"W$J$;::"@11V27H@a=@D2B@"B$(1)"@17V31H@b="B$(2)"@10V34H@c=@D35H@"B$(3):B$(1)"??"Ē1:207,59207,131:3:207,131269,131207,59:B$(2)"??"Ē1:207,131269,131:3:207,59207,131:269,131207,59:208,59208,131,75D,43I,43:I1,43I1,75D1,43I1,43:Y9I,75I,90I38,90I,75:I1,75I1,90:w935,I73,I73,I1535,I:9I,43I38,43I38,59I,43:935,I35,I3217,I3235,I:9I,90I18,90I18,58I,90:*:123:179,32179,159:180,32180,:\8|3:H117:H273:V143:V290:34:H135:34:H117:V275:34:35,4317,76:35,7673,90:~8}H1101:H2157:V143:V290:348~6:I1934:I,76I,89::85:I3771:I,44I,74::3:81:I1934:I,44I,74::I3771:I,76I,89::,9I,43Icimal places.":118F7xFO1:"@11V1H@"TY$"@D1H@The answer is "J(2)"."7y36262,(36262)FO:27:HT1:VT7:BL25:LN13:29:HT27:BL12:29::31:(QQ)01447z488{"@I4V26H@(SAMPLE) @D26H@ X X@G@2@R@ @UL@"W$"@5V37H@X @I5V@":I1124:B$(I):113L6uFO1:"@11V1H@"TY$"@D1H@The answer is "L"@LU@"W$J$J$"@D3B@"U"@21H@."6v165:HT10:VT10:MX8:L1:36:J(2)((AN(M(P)))102.5)(102.5):TU$(J(2)):J(2)(TU$):INJ(2)ĺ"@11V1H@"C$:1217w2000:(IN$)TP2ĺ"@11V1H@Wrong number@D1H@of dert.";5qHT10:VT10:MX2:L1:36:ANIN:HT15:MX4:36:152v5r"@11V1H@"24)"@D1H@"24):ANLINUĺ"@11V1H@"C$:1185sAE0:I120:DAN(M(P))(II):D(D)LI:UD:LANINUAE16t:AEĺ"@11V1H@"C$" but reenter in@D1H@simplest form.@10V15H@"4):1H@What is the length in@D1H@simplest RADICAL FORM?@L13H9V@"W$J$J$J$5p"@10V12H@*@14V1H@Use the "D$" keys to@D1H@fill in your answer where@D1H@the lighted cursor shows.@D1H@Use "(91)(2)"] to backspace and@D1H@"(91)"RETURN] when you finish@D1H@each pa = "L"@L16V11H@"W$J$J$"@D13H@"G;v3m"@17H@(radical)":35:B$((B)):B$(B$,1,5):"@D4H@??? = "B$"@17H@(decimal)":93n3:I$:140:I$:P195((QQ)0):FO0:30D4oI19:M(I)R(3)::141:I19:B$(I)(B(I))::B$(M(P))"??":150:TR111:136:"@7V:(D$(3))D$(3)B$(3)2k"@11V3H@"D$(1)" = "D$(2)" "(432(M(P)3))" "D$(3):35:B(D$(2))(D$(3))2(D$(3))(M(P)3):"@13V3H@"D$(1)" = "B:35:I120:DB(II):D(D)LI:GD 3l:"@L14V8H@"W$J$"@15V4H@?? =@10H@"B;:L1G1109:"@17V4H@??"::"@7V3H@"B$(1)" + "B$(2)" = "B$(3)1i35:I13:B$(I)(((B(I)2).2))::B$(M(P))"(??)@G@2@R@":"@9V3H@"B$(1)" + "B$(2)" = "B$(3):35:D$(1)B$(M(P)):D$(2)B$(2):D$(3)B$(32(B$(3)D$(1))):I13:(B$(I))(D$(2))D$(3)D$(2):D$(2)B$(I)2jght "H$", so we can@D1H@solve:":I13:B$(I)(B(I))::B$(M(P))"??":"@14V3H@("B$(1)")@G@2@R@ + ("B$(2)")@G@2";&1h"@R@ = ("B$(3)")@G@2@R16V1H@to find the length of the@D1H@third side.":136:27:HT1:VT7:BL25:LN13:29:I13:B$(I)"("B$(I)")@G@2@R@c@15C14H@+@23H@=@R11H@2@20H@2@29H@2":9C/eP448:P1103:M(1)3/fM(2)R(3):M(3)R(3):M(4)R(3):M(2)M(3)M(2)M(4)M(3)M(4)102:141p0g140:"@2D1H@We know that a@G@2@R@ + b@G@2@R@ = c@G@2@RD1H@when c is the length of@D1H@the "F$" of a@D1H@ri48:H19:H2270:V135:V2131:34:"@5V2H@For any right "H$" with side@D2H@lengths a,b, and c where c is the@D2H@length of the "F$", the@D2H@relationship among the lengths of"'/d"@2H@the sides is given by the@D2H@expression:@L133C13V9H@a@18H138C@b@27H5C@"F$" (the side c,@D1H@opposite the right angle)@D1H@in comparison to the@D1H@lengths of the other two@D1H@sides.":35-b"@D1H@We refer to this as the@D1H@Pythagorean Theorem,@D1H@named after the Greek@D1H@philosopher who proved it@D1H@true.":9.cP1HL5C@= c@15CR@2@15C19H7V@C":9,`5:206,131269,131:6:206,59206,131:2:206,59269,131:3:206,123215,123215,131:"@28H12V@a@34HU@c@6D2B@b@5V1H@The equation a@G@2@R@ + b@G@2@R@ = c@G@2@R@"u-a"@1H@describes the length of@D1H@ANY right "H$"'s@D1H@does not@D25H@change the@D25H@total area,@D25H@but now the@D25H@area a@G@2@R@ + b@G@2@RD25H@is combined in",_"@25H@a single@D25H@"S$" bounded@D25H@by the long@D25H@sides of the@D25H@"H$"s.@L16V2H133C@a@15CR@2":35:"@16V6HL138C@+ b@15CR@2":35:"@16V14I103118:I,DI,44:DD1.8::D44:I121155:I,44I,D:DD.4::D77:I103137:I,DI,89:DD.4::D90:I140155:I,DI,89:DD1.8::2:H1103:H2138:V175:V289k+^H1,V11H2,V2:H1H1.6:H2H2.6:V1V11:V2V21:V14594:126:124:127:"@9V25H@This ":I75435:131:124:0:131:3::I351198:132:124:0:132:3::II4:132:125:120,44101,75:"@13H6V@b@3BD@b":I43585:133:124:0:133:3::I171438:134:124:0*]134:3::II7:134:102,76139,90:125:"@10V13H@a@3U3B@b":128:D73:H1101:H2157:V143:V290:34:"@4V15H@a@3F@b@3F2D@a@B4D@b@2D3B@a@5B@b@5B2U@a@B4U@b":HT2:VT5:BL9:LN7:29:124:I17988:DI18:129:124:0:129:3::II4:DD4:129:"@13H6V@b@D3B@b":I351018:130:124:0:130:3:)\II6:130:"@13H10V@a@3B@a"S$"@D13H@has an area of b@G@2@R@.":35:128:3'Z35,4317,76:35,7673,90:"@16V1H@Each pair of "H$"s has a combined@D1H@area of a*b.":27:HT13:VT4:BL26:LN11:29:"@16V1H@"38)"@D1H@"20)"@5V25H@Here we move@D25H@the identical@D25H@"H$"s."([35:as@D13H@been divided into measured@D13H@sections.":124:"@4V3H@a@7H@b@3F3D@b@3DB@a@2D4B@b@6B@a@2U3B@a@3UB@b":35:126/'Y"@9V13H@The area of the small@D13H@"S$" is a@G@2@R@, since each@D13H@of its sides is 'a' long.":35:127:"@13V13H@The large inside "@D1H@roots of those "D$"s@D1H@are "Y" and "Y1", so we may@D1H@"Z$"e that "X"'s@D1H@"E$" is not an@D1H@"Y$", and falls@D1H@between "Y" and "Y1".":9%U24576,28:3:O$"RUNAM5.3-3"%VM87,99,101,110%WP88,96,48v&X"@5V13H@Here is a "S$" which h)11:AYY:B(Y1)(Y1):XR(BA1)A:J$"@3BD@"X:35:"@8V1H@"X" is within the range@D1H@of our table, but it@D1H@falls between the "V$"s@D1H@"A" and "B" in the X@G@2@RD1H@column.@I29H"5Y"V@"A"@D29H@"B%T35:"@26H"5Y"V@"Y"@D26H@"Y1"@I10H12V@The "S$RYR(12)11:XYY:J$"@3BD@"X:35:"@8V1H@"X" appears in the X@G@2@R2D1H@column of the table.@I29H"5Y"V@"X:35:"@U26H@"Y"@I23H10V@We@2D1H@need merely read the@2D1H@corresponding "V$" in":"@D1H@the X column to find "Y",@2D1H@"X"'s "E$".":9$SYR(12ich have "E$"s@D1H@of "A$(X)" and "A$(X1)".@I"X5"V26H@"X"@5F@"A$(X)"@D26H@"X1"@5F@"A$(X1)I$:35:"@13V1H@We may "Z$"e or@D1H@approximate to find a@D1H@"V$" for "XZ"'s "S$"Q"@1H@root, somewhere in "V$"@D1H@between ";A$(X)" and@D1H@"A$(X1)".":9#@the "E$" of "U","O!N"@1H@which is "(A$(X))10".@I13H5V@= "(A$(X))10I$:9!OJ$J$J$:XR(13)10:Z(1):Z(Z102.5)(102.5):"@5V8H@"XZ:35:"@7V1H@"XZ" is uithin the@D1H@range of our table, but@D1H@falls between "X" and "X1",""P"@1H@wh00) does appear@D1H@and has a "E$" of@D1H@"A$(X)"."!M"@I"X5"V26H@"X"@5F@"A$(X)I$:35:"@10H11V@Since we used@D1H@100 to the FIRST power@D1H@as a divisor to fit the@D1H@range of the table, we@D1H@multiply "A$(X)" by 10 to@D1H@the FIRST power to get@D1H1H@Read the "V$" of its@2D1H@"E$" in the @LU@"W$"@DB@X@2D1H@column. The answer is@2D1HI@"A$(X)"@5V11H@= "A$(X)"@"X5"V33H@"A$(X)"@I16V7H@.":93 LXR(12)11:UX100:J$J$"@D8H@"U:35:"@7V1H@"U" falls outside the@D1H@range of our table, but@D1H@("U" / 1ween the "V$"s@D6H@you can find in the table you@D6H@must "Z$"e or use the@D6H@approximation technique.@G9V23H@2@L10V34H@"W$"@36H11V@X":9JP548:135:P75,76,79,82,83KXR(12)11:"@D2B@"X"@8V1H@Locate "X" in the table.@"X5"V26HI@"X:35:"@I10Vnts menu of this@D6H@program).":9H34:V2155:34:"@5V2H@To find the "E$" of X in a@D2H@table, using the second method:@3D2H@<1> Locate X in the X column of the@2D6H@table and read your desired@D6H@"V$" in the X column."I"@D2H@<2> If X falls bett"F"@6H@for each negative "Y$" power.@L20H13V@"W$"@D22H@X":27:29:"@9V2H@<3> If X falls between "V$"s you@D6H@can find in the table you must@D6H@"Z$"e or use the""G"@6H@approximation technique (see@D6H@'Approximate Square Roots' in@D6H@the ConteX is outside the range@D6H@covered in the table, use the"E"@6H@"V$" of X times any "Y$"@D6H@power of 100 instead.@2D6H@When you read move the@D6H@decimal point one position right@D6H@for each positive "Y$" power@D6H@to which you raised 100, or lef" of X in a@D2H@table, using the first method:@3D2H@<1> Locate the "V$" of X in the X@2D6H@column of the table and read the@2D6H@given "V$" for"9D"@L12V21H@"W$"@31HL@"W$"@13V23H@X in the@33H@X@2D6H@column.":27:HT2:VT9:BL36:LN10:29:"@9V2H@<2> If A"@1H@is 10Y, and the "E$" of .01X is@D1H@.1Y.@D2H@This fact helps us find the@D1H@"E$"s of "D$"s which are@D1H@outside the range of the table of@D1H@"S$"s and "E$"s.":9BP67,72,48CH19:H2270:V135:V2155:34:V259:34:"@5V2H@To find the "E$40?@2D4HI@PRESS "(91)"Y] FOR YES OR "(91)"N] FOR NO."I$@14:KK128:K89K7864:"@8V36H@"(K)"@2D1H@The answer is NO. If we "S$" 40 we@D1H@get 1600. In fact, it is always true@D1H@that for any "D$" X which has a@D1H@"E$" Y, the "E$" of 100X"$"s which do@D1H@not appear in the table@D1H@can be estimated by@D1H@"Z$"ion (fitting@D1H@between) or by using@D1H@multiplication.":99?"@4V1H@We saw in the table that the "S$"@D1H@root of 16 is 4.0000 (= 4). Does that@D1H@mean that the "E$" of 160 = e@D1H@"D$" line?).":27:HT1:VT4:BL24:LN16:29="@5V1H@SECOND - the "E$"@D1H@"V$"s are rounded off.@D1H@Since any "E$"@D1H@which is @I@not an "Y$"@D1HI@is irrational, it@D1HI@cannot be shown fully@I@ in@D1H@decimal form.":27>"@13V1H@THIRD - "Vd "S$"@D1H@roots.":35:"@10V1H@FIRST - This table shows@D1H@us "V$"s which@D1H@correspond to "Y$1<"@1H@"V$"s of X, but there@D1H@is an infinite "D$" of@D1H@intermediate "V$"s@D1H@between the "V$"s in@D1H@each column (remember@D1H@how they looked on th:123:"@6V1H@This is a sample from@2D1H@such a table, with the@2D1H@control column (the@2D1H@middle of the three@2D1H@"D$" lines) on the@2D1H@left.":9a;123:"@4V1H@There are three very@D1H@important points to@D1H@learn about tables of@D1H@"S$"s anline.":27:"@16V2H@If we take a cross section of all "9"@2H@three lines and list the "V$"s, we@D2H@have a "S$"s and roots table. @5V11H@"26)"@4D11H@"26)"@13V11H@"26):5:I522567:I,32I,64:I1,32I1,64:I,72I,104:I1,72I1,104::3:9:"@E12V6H@"R$"@11H@"R$"@16H@"R$"@24H@"R$"@33H@"R$"@15CE13V6H@0@4FI@"O$"16@2F@"O$"81@5F@"O$"256@5F@"O$"625"I$88"@16V2H@Notice the third line.":35:"@13V3H@X@G@2@R16V26H@Each "D$"@D2H@on the third line is the "S$" of@D2H@its counterpart on the middle "25@I6H5V@0@4DB@0":35:"@L2H4V@"W$"@BD@X@9V3H@X@16V2H@Notice that each "D$" on the top"61:H19:H2270:V1123:V2155:34:3:"@2H@line is the "E$" of another@D2H@"D$" on the lower line.":27:HT2:VT16:BL36:LN3:29n7I537:"@12V"I"H133C@"(10):4,853P52,58,59,63,484I537:"@133C4V"I"H@"(10)"@8VB@"(10)::"@E4V6H@"R$"@4DB@"R$"@4F@"R$"@4UB@"R$"@4F@"R$"@4DB@"R$"@7F@"R$"@4UB@"R$"@8F@"R$"@4DB@"R$"@E15C@"X5"@I5V11H@"O$"2@3F@"O$"3@6F@"O$"4@7F@"O$"5@9V11H@"O$"4@3F@"O$"9@6F@"O$"16@6F@"O$P0:30:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@2D10H@<3> EXAMPLE@10H2D@<4> SAMPLE PROBLEM@10H3D@<0> RETURN TO CONTENTS@I10H6D@WHICH (0-4) ??"I$:4:Kİ32:3:O$"RUNAM5.3"1MK:P1:36320M,(36320M)1:30:312C486:M51,66,7":Y$"integer":H$"triangle":F$"hypotenuse":J$(12):Z$"interpolat":H16368:QQ36251:(QQ)0110-I1124:A$(((I)104.5)(104.5)):I16A$"4.0000".(A$)6A$A$"0":46/A$(I)A$:B$(I)"@26H@"(I)R$((I2.2))R$A$:0M0:36:IN$"":I0PS:IN$IN$((512I))::IN(IN$):"@R@": +35339:I$"@I@":TY$"WRONG.":C$"CORRECT.":R(X)((1)X)1:X(X)(1)X:O$(4):A$(25),B$(25):R$(9):E$"square root":W$(16)"@R@"(12):C(24576)d,D$"number":S$"square":V$"valueSMXĺ"@"HTPSL"HI@"((512PS))I$;n &21:(PSMX)(K141)(K136)38:PSMXĺ"@"HTPSL"H@"((512PS)); 'K14142:K136PSĹ511PS,32:PSPS1 (KK128:K47K58K45K46ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX42 )37: *(PS0)(P10)"@2HD@"M:3:" 31051:33- 30976J !"@21V1HLI@"19)"@RI@": "H11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2: #I11500::K(16384):MK15515: $35400,L(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32::H,0$ %P22:( PP(K149)(P0)(K136):252 :24< 145X K13632:K14931:50 "@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINUE"I$:H,0 14:K16028:"@22V1HI@"36)I$: "@R0K"VT"V@";:I1LN:"@"HT"H@"BL)"":: "@2H1V@"C"@5H@"P810:K21PP1:310 K8PP1:32:P48< 30:50U ZH(1):21:(QQ)ı (P(K136K149)(M4))5:K155ı "@40X40YN@";:21:K155K205ı H,0:19 :18 145 32:K15548:144 H,0 K(16384):K128724577:30000:166!4300:1002:M14:KK176:K0K44:U7^:6h145p11 "@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE"I$ 14 KK128:K27(QQ)16:K21K                                     S,THE PYTHAGOREAN THEOREM,FINDING SQUARE ROOTS TEST,F=(222)253Ħ=::3::"++ERROR++";::(222):"AT LINE #"(218)(219)256:"IN AM5.3":>>>4G"V"H"H@"R$(T)"@"H1"H@"X$(T):TT1::42:"@13V"H"H@"R$(X);:42:"@"H"H@"O):TX:G1X:"@"13G"V"H"H@"R$(T)"@"H1"H@"X$(T):TT1::42::<XX335:RD4:S18:ENS:IN((1)50):136::4=APPROXIMATE SQUARE ROOTS,COMPUTING SQUARE ROOTS,USING TABLERO$(I)"":SQ$(I)""v;:I16:(RO$(I))2RD2(RD3)(RO$(I))RO$(I)RO$(I)"0":RD2RO$(I)(RO$(I),1,2)".0";:I16:"@"H1"H@"RO$(I)"@"H2"H@"SQ$(I)SP)"@D@";::ENEN1:<X13:"@13V"H"H@"R$(7X):42:"@13V"H"H@"O):T7X:G1X:"@"13R3R)):R(CC3):RO$(2)(RO(CCR2R)):SQ$(2)(SQ(CCR2R)):R(CC2):RO$(3)(RO(CCR1R)):SQ$(3)(SQ(CCR1R)):RO$(4)HH$:SQ$(4)(XX);RO$(5)(RO(CC1)):SQ$(5)(SQ(CC1)):RO$(6)(RO(CC2)):SQ$(6)(SQ(CC2)):"@9V@";:I16:(RO$(I))I49RO(EN)IN:SQ(EN)ININ:SQ(EN)(SQ(EN)10RD.5)(10RD.5):I1EN1:JI1EN:RO(I)RO(J)TPRO(I):T1SQ(I):RO(I)RO(J):SQ(I)SQ(J):RO(J)TP:SQ(J)T1:JI9J,I:CC0:I1EN:SQ(I)XXCCI::R(CC4):RO$(1)(RO(CCR3R)):SQ$(1)(SQ(CCR(2):X$((XX)):X(X$):((X$,4,1))9((X$,5,1))9((X$,6,1))9X(X)XX170129:f8O138(X$(O))OPX$(O)X$(O)" ":1318:8"@8V21HI@ ROOT SQUARE@I@":191,72191,120:192,72192,120:8OT$(OT):(OT$)5OT$OT$"0"9re root of "XX"@D2H@to "A" decimal "CD$".@I@":R7~14:KK128:K89K78126:7"@I8V27H@ROOT SQUARE@13V33HI@"XX:220,64220,145:221,64221,145:7"@I8V21H@ ROOT SQUARE @I13V29H@"XX:192,64192,145:193,64193,145:[8XXR(202)112:A:"@16V2H@"W$" The answer is "RO(CC(AN2)(AN1))" because@D2H@its square is closest to "XX"."6{36259,(36259)(RE):27:37::(36251)57:3:(4)"RUNAM5.3-3"6|CD$"place":A1CD$CD$"s"/7}"@5V2HI@"36)"@D2H@"36)"@5V2H@Approximate the squaf "XX"?":T$(1)"@21H11V@"(RO(CC1)):"@11V28H@"SQ(CC1)"@2D28H@"SQ(CC1)5xT$(2)"@13V21H@"(RO(CC1)):"@12V28H@"XX;T$(1)T$(2):41:LN2:30:AN1:(XXSQ(CC1))(XXSQ(CC1))AN25yLN4:HT2:VT15:INBL:BL36:29:INANĺ"@17V2H@"C$:123_6zRE1@thousandth.":AN3R$"You need to end approximation."J4u"@16V2H@"W$R$t4v27:29:VT8:LN7:BL19:29:AN1113J5wHT21:VT9:BL6:LN6:29:HT28:BL10:29:"@8V2H@Which of your@D2H@approximations is@D2H@the best estimate@D2H@of the square root@D2H@oO(CC1)RO(CC1)).0018AN33sHT2:VT8:BL19:LN8:29:VT15:BL36:LN4:29:41:"@8V2H@What will you do@D2H@next?@2D@";:I13:"@2H@"T$(I)::LN3:30:INBL:HT2:VT15:BL36:LN4:29:INANĺ"@16V2H@"C$:11874tRE1:R$"You need to select another@D2HT$:HT26:VT12:MX1:L1:35:44:H121:H228:INOTIN1000:I1EN:INRO(I)ĺ"@12V2HI@You've tried @D2H@that already.@I@":IEN1::1133r:136:T$(1)"@11V2H@Select a thousandth":T$(2)"@12V2H@Select a ten@D2H@thousandth":T$(3)"@14V2H@"T$(3):AN1:(R9:AN1105:AN3119:HT21:VT9:LN6:BL6:29:HT28:BL10:29:OTRO(CC1):134:I1EN:RO(I)0:SQ(I)0::RO(1)(XX):SQ(1)XX:RD6:EN22qHH$" ":SP0:"@12V28H@"XX"@8V2H@Select a thousandth@D2H@to approximate the@D2H@square root of "XX":@12V21H@"OBL:HT2:VT15:BL36:LN4:29:INANĺ"@16V2H@"C$:1120mRE1:R$"You need to select another@D2H@hundredth.":AN2R$"You need to select a@D2H@thousandth."0nAN3R$"You should end approximation."0o"@16V2H@"W$" "R$1p27:29:VT8:LN7:BL19:2@I@":105y/k136:T$(1)"Select a hundredth":T$(2)"Select a thousandth":AN1:(RO(CC1)RO(CC1)).018AN2:A2AN360lHT2:VT8:BL19:LN8:29:VT15:BL36:LN4:29:41:"@8V2H@What will you do@D2H@next?@2D@";:I13:"@2H@"T$(I)::V111:LN3:30:IN hundredth@D2H@to approximate the@D2H@square root of "XX":@12V21H@"OT$:HT25:VT12:MX2:L1:35:44:H121:H229:INOTIN100:I1EN:INRO(I)ĺ"@12V2HI@You've tried @D2H@that already.@I@":IEN1::105 /j:PS1ĺ"@I12V2H@One digit @D2H@only! "W$" "R$-g27:29:VT8:LN7:BL19:29:AN196:AN3119:HT22:VT9:LN6:BL5:29:HT28:BL10:29:OTRO(CC1):I1EN:RO(I)0:SQ(I)0::RO(1)(XX):SQ(1)XX:RD4:EN2-hHH$" ":OT$(OT):(OT$)4OT$OT$".0".iSP1:"@12V29H@"XX"@8V2H@Select aD@";:I13:"@2H@"T$(I)::V111:LN3:30:INBL:HT2:VT15:BL36:LN4:29:INANĺ"@16V2H@"C$:103,dRE1:R$"You need to select another@D2H@tenth.":AN2R$"You need to select a@D2H@hundredth.",eAN3R$"You should end approximation."-f"@16V2H@a:PS1ĺ"@I12V2H@Must be a @D2H@tenth. @I@":96+b136:T$(1)"Select a tenth":T$(2)"Select a hundredth":AN1:(RO(CC1)RO(CC1)).18AN2:A1AN3b,cHT2:VT8:BL19:LN8:29:VT15:BL36:LN4:29:41:"@8V2H@What will you do@D2H@next?@2XX:RD2:EN2*`"@12V30H@"XX"@8V2H@Select a tenth@D2H@to approximate the@D2H@square root of "XX":":SP2:"@12V22H@"OT".":HT25:VT12:MX2:L1:35:44:H122:H230:INOTIN10:I1EN:INRO(I)ĺ"@12V2HI@You've tried @D2H@that already.@I@":IEN1::96;+"You need to select another@D2H@integer.":RO(CC1)1RO(CC1)R$"You need to select a tenth."v)^"@16V2H@"W$" "R$ *_27:29:VT8:LN7:BL19:29:AN189:HT22:VT9:LN6:BL5:29:HT28:BL10:29:OTRO(CC1):I1EN:RO(I)0:SQ(I)0::RO(1)(XX):SQ(1)T$(3)"End approximation":AN1:RO(CC1)1RO(CC1)AN2(\HT2:VT8:BL19:LN8:29:VT15:BL36:LN4:29:41:"@8V2H@What will you do@D2H@next?@2D@";:I13:"@2H@"T$(I)::V111:LN3:30:INBL:HT2:VT15:BL36:LN4:29:INANĺ"@16V2H@"C$:95`)]RE1:R$ "XX":":HT23:VT12:L1:MX2:35:44:H123:H231:RD0:I1EN:INRO(I)ĺ"@12V2HI@You've tried @D2H@that already.@I@":IEN1::89'ZRE0::PS2ĺ"@I12V2H@Try a 2-digit@D2H@number @I@":898([136:T$(1)"Select an integer":T$(2)"Select a tenth":9&XP195((36251)0):36:H19:H2270:V135:V2155:40:V259:40:129:124:I19:RO(I)0:SQ(I)0:RO$(I)"":SQ$(I)""::EN2:133:SQ(1)XX:RO(1)(XX):SP3:HH$" "'Y"@12V31H@"XX"@8V2H@Select an integer@D2H@to approximate the@D2H@square root of(10A.5):HT2:VT8:LN11:BL36:29:"@9V2H@By approximation, we now know that@2D2H@the square root of "XX" is "WE" (to@2D2H@"A" decimal "Z$"). We could find a@2D2H@more precise solution by continuing@2D2H@approximation";&W" to smaller intervals.":24:RR(Y)D1I1000:YY1:$UHT21:BL17:29:"@16V6HI@thousandths@I@ ":128:O6:I16:R$(I)(RR(I)):R$(I)(R$(I),1,6):X$(I)(((R$(I))2),1,10):I:H128:142:"@I12V21H@"R$(4)"@F@"BO$X$(4)"@2D21H@"R$(3)"@F@"BO$X$(3)I$:27%VWE((XX)10A.5)@D6HI@hundredths@D6HI@place.":128:Y1:I24:RR(Y)D1I100:YY1::I16:R$(I)(RR(I)):R$(I)(R$(I),1,5):X$(I)(((R$(I))2),1,8)::H21:H129:O6:142$T"@I12V21H@"R$(4)"@2F@"BO$"@F@"X$(4)"@2D21H@"R$(3)"@2F@"BO$"@F@"X$(3)I$:27:A286:Y1:I),1,4):X$(I)(((R$(I))2),1,6)::O4:OP6:130"RH27:H132:142:"@I12V27H@"R$(4)"@F@"X$(4)"@2D27H@"R$(3)"@F@"X$(3)I$:27:A186:29:"@9V2H@<3> Continue until@D6H@the desired"#S"@6H@number of@D6H@decimal places@D6H@is reached.@2D6H@Here, to thepeat the process@D6H@from step <1> using@D6H@tenths between the@D6H@consecutive integers":Y1:I24:RR(Y)D1I10:YY1:1"Q"@6H@until you find@D6H@consecutive tenths@D6H@which are too large@D6H@and too small.":127:I16:R$(I)(RR(I)):R$(I)(R$(Ind@D6H@smaller integers@D6H@until you have two@D6H@consecutive numbers,@D6H@one too large, the@D6H@other too small.":42:42:127:O2:H28:H133:142|!P"@12V28HI@"R$(4)"@33H@"X$(4)"@2D28H@"R$(3)"@33H@"X$(3)I$:27:HT2:VT8:BL36:LN11:29:"@9V2H@<2> Re"places":A1Z$"place"NH19:H2270:V135:V2155:40:V259:40:"@5V2H@Approximate the square root of "XX"@D2H@to "A" decimal "Z$".@9V2H@<1> Square an integer@D6H@which you think@D6H@may be close to the@D6H@square root of "XX"." O"@6H@Try larger aH@until you reach the desired@D6H@degree of accuracy for your@D6H@estimate of the root of X.":9LP457:129:RR(XX):DRR1:Y1:I33:RR(Y)DI:YY1::I16:R$(I)(RR(I)):R$(I)(R$(I),1,2):X$(I)(((R$(I))2),1,3)::P1A3MOP3:130:Z$d consecutive@D6H@integers which are too small and@D6H@too large, respectively, repeat@D6H@the process using tenths between@D6H@the two integers."`K27:HT2:VT7:BL36:LN12:29:"@8V2H@<3> Continue using ever smaller@D6H@increments for approximation@D6are root of X:@2D2H@<1> Square an integer which you@D6H@guess will result in a number@D6H@close to X. If the result is@D6H@too small, try a larger integer.J"@6H@If the result is too large, try@D6H@a smaller integer.":42:"@D2H@<2> When you have founHI@"T$(4)I$:A$W$:BL4A$C$GHT2:VT16:BL36:LN3:29:"@16V2H@"A$" The square of 25.884 is@D2H@closer to 670 than is the square of@D2H@any other number shown.":9HP157IH19:H2270:V134:V2156:40:V152:40:"@5V2H@To approximate the squuare root@D2H@of 670?@I21H8V@ ROOT SQUARE "I$:T$(1)"25.8@3F@665.64":T$(2)"25.9@3F@670.81":T$(3)"25.885@F@670.033226":T$(4)"25.884@F@669.981456":T$(5)"25.883@F@699.929689":"@10V21H@";:I15FT$(I)"@D21H@";:I:41:HT21:V110:LN5:30:"@13V21 we want.":9DH19:V135:H2270:V259:40:V1156:40:192,72192,120:193,72193,120:"@5V2H@Here is a list of numbers which are@D2H@close to the square root of 670.@9V2H@Which of these is@D2H@the best guess, or@D2H@APPROXIMATION, for"E"@2H@the sqo "RD$R$R$"@2BD@39 must be@D2H@between 6 and 7.@G12V3H@2@15H@2@RD@":42:"@D2H@(6.2) = 38.44 and (6.3) = 39.69," C"@2H@so it must be between 6.2 and 6.3.@G15V7H@2@26H@2@RD@":42:"@D2H@We can compare closer and closer@D2H@numbers to be as accurate asH19:V135:H2270:V275:40:"@5V2H@We can APPROXIMATE the square root@D2H@of a number X by finding numbers@D2H@which have squares slightly smaller@D2H@and larger than X.":42:"@10V2H@What is the square root of 39?":42rB"@12V2H@6 = 36 and 7 = 49, s01:203,118253,118:207,98207,140:203,135253,135@27:0:I210247:I,102I,134::5:248,99248,144:3:"@15V2H@The sides of a square with@D2H@an area of 6 would be@D2H@longer than 2, but shorter@D2H@than 3.":"@11V31H@? ? ? ?@2DB@?@2DB@?":9Aimensions of a square@D2H@when we already know its area.@9V2H@"A$4?1:I8414717:203,I272,I:I:I20626921:I,80I,140:I1,80I1,140::27:"@12V2H@"A$"@27HU@4@D12H@2":0:I208268:I,85I,134::5:227,98227,140:249,98249,140:203,101253,1:MK:P1:36320M,(36320M)1:36:37F;C60,3000,4000,5000,6000Y<M61,72,76,88l=P62,65,68,57A>H19:V135:H2270:V267:40:A$"A square with an area of 9@D2H@has sides 3 units long.":"@5V2H@Finding a square root is like@D2H@finding the dM5.3-3"18C5Ĺ36251,3:I15:36258I,0::889M0:P0:36:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@11V10H@<3> EXAMPLE@10H13V@<4> SAMPLE PROBLEM@10H16V@<0> RETURN TO CONTENTS":"@I12H22V@WHICH (0-4) ??"I$:MN0:MX4:4:Kİ38:52(4820,14242,14249,14842,15420,154:"@18V3H@<0>":59,3259,159:60,3260,159:I115:36251I,0::MN0:MX5:4:CK:36:24(C$(K))2:"@2VI@"C$(K)"@I@":Cİ38:3:(4)"RUNALGEBRA 5"637:24576,C:C3C4İ3:(4)"RUNAM5.3-2"7C2İ3:(4)"RUNAI"> "C$(I):::"@10H18V@<0> RETURN TO ALGEBRA MENU@2V7HI@"31)"@12H22V@WHICH (0-5) ??"I$:P0:M0:C0:36:H117:H245:I04:V14516I:V2V112:40:"@3H"62I"V@<"I1">":V14516(I1)16(I4)532,V232,V136,V13:31,V231,V127,V13::20,15413,1R@":335339:C$(5):I15:C$(I):I:I$"@I@":W$"WRONG.":C$"CORRECT.":R(X)((1)X)1:X(X)(1)X:N$(2)","(1):A$(20),B$(20),RO(30),SQ(30):RD$"@UL@"(16)"@R@":R$(12):BO$"@I@ @I10B@"4"@20H5V@CONTENTS@6V@":I15:"@10H@<"(K141)(K136)46:PSMXĺ"@"HTPSL"H@"((512PS));d/K14150:K136PSĹ511PS,32:PSPS10KK128:K47K58K45K46ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX501452(PS0)44:IN$"":I0PS:IN$IN$((512I))::IN(IN$):"@oice, and the "(91)"RETURN] key to @D2H@select your choice.":i*I11500::K(16384):MK15515:z+I1500::,35400,L(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32::16368,0-PSMXĺ"@"HTPSL"HI@"((512PS))I$;9.21:(PSMX) you finish each part.":C $"@2H1V@"C"@5H@"P(P10)"@2HD@"M:3:R %31051:39] &30976z '"@21V1HLI@"19)"@RI@": (H11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2:?)"@16V2H@Use "(91)(2)","(1)"] keys to light the correct@D2H@chBLBL1:VV(GH1C1M4(T$(BL))1):BLBL(GH1C1M4(T$(BL))1):VV1VV2:BLLN` "31 #"@15V2H@Use the number keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Use "(91)(2)"] to backspace and@D2H@"(91)"RETURN] whenT"V@";:I1LN:"@"HT"H@"BL)""::< BL1:VV1:V2V1LN1 "@"V"V"HT"HI@"T$(BL)I$:21:KYK:16368,0:"@"V"V"HT"H@"T$(BL):KY141ı KY149VV1:BLBL1:VV(GH1C1M4(T$(BL))1):BLBL(GH1C1M4(T$(BL))1):VV230X !KY136VV1:ALGEBRA 5": 16368,06 K(16384):K12822:Y PP(K149)(P0)(K136):25c :24m 145 K13638:K14937:59 "@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINUE@I@":16368,0 14:K16028:"@22V1HI@"36)"@I@":# "@R0K"V6:K21K810:K21PP1:37: K8PP1:38:P57F 36:59b ZH(1):21:(36251)ı (P(K136K149)(M4))5:K155ı "@40X40YN@";:21:K155(K205C)ı 16368,0:19 :18 145 38:K15557:3:(4)"RUN 524577:30000:145!5140:1002:::S14:KK176:KMNKMX4:[7d:6n145v11 "@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE"I$ 14 KK128:K27(36251)01                        |>>"$256:"IN AM5.2-3":"**M3,M4 ALGS#~İ50310:T1TP:50310:T2TP:50310:T3TP:50310:T4TP:T1(1T3)50302:T5T1T3:T6T5T2:T6(T6)50302:T6T6T6:T69950302:#TPD(9)1(D(2)1):#(222)253Ħ$:26:::"++ERROR++";::(222):"AT LINE #"(218)(219)H@ 1 @12V2H@What is the solution? X=":VT12:HT26:620:"@L15V2H@"18)"@D2H@"18)"@R15V2H@"S1$(INT6)"X = "T6:TX(INT6)TX"@340:NCNCTX::410:"pPM4:P1:NC0:5410:36258,NC:26:Cĺ(4)"RUNALGEBRA 5""z(4)"RUNAM5.3-2""Pò***C4 ALGS #|IJ*elow. "I$:50302I!%"@6V26H@"T1"="T2"@U@"RD$"@DB@X"(452(T30));(T3)!("@10V2H@How many solutions are there?":HT32:VT10:L1:MX3:620:T9((T2T5)1):"@L15V2H@"18)"@D2H@"18)"@R15V2H@"S1$(INT9)S$(T9):TX(INT9):T95440:340:365{","@10V32ere are no solutions.":S$(1)"there is one solution.":P145(M4):365:"@L5V24H@"7)"@R10V2H@"36)"@12V2H@"36):400!$V136:V276:500:H2165:500:"@I5V2H@Solve the radical"4)"@D2H@equation on the"6)"@D2H@right, then answer @D2H@the questions b =@"4(T20)"F@"T6" ?":HT15(T10)(T20):I02:"@"15I"V"HT"H@"T6"@D2B@ ":I51400:I5,I:"@15V8H@"15)TP((T2T5)0):"@17V2H@"S$(TP)", so "T6"@D2H@is "T$(TP)"a solution.":3105410:1200x "S1$(0)"@I@WRONG@I@, ":S1$(1)C$", ":S$(0)"thSolve@I@ (simplify)@D6H@the equation.""@16V28H@"T6"=X":340:HT2:VT8:BL36:LN11:350:"@9V2H@<4> @I@Check@I@ the solution(s) in the@D6H@original equation.":600:"@13V8H@"T1" = "T2"@U@"RD$R$"@D2B@X "(432(T30));(T3):600m"@15V"8(T10)"H@X@radical term.@26H@"T1;(452(T30));(T3)"="T2"@U@"RD$"@DB@X":600:"@12V2H@<2> @I@Raise@I@ both sides"&"@6H@to the 'nth'@D6H@power.@U23H@("T1;(452(T30));(T3)")@G@2@R@="T2"@G@2@R@X":600:"@14V27H@"T5"@G@2@R@="T2"@G@2@R@X":600:"@16V2H@<3> @I@iginal equation.":310aP41200:S$(0)"@I@TRUE@I@":S$(1)"@I@FALSE@I@":T$(0)"":T$(1)"NOT "V136:V260:502:50302:"@5V2H@Solve the@D2H@equation: "T1"="T2"@U@"RD$"@DB@X"(432(T30));(T3):V2156:500:600i"@9V2H@<1> @I@Isolate@I@ the@D6HRaise@I@ both sides of the"\"@6H@equation to the nth power (n is@D6H@the index of the radical.)":600:"@14V2H@<3> @I@Solve@I@ the resulting equation.":600^"@16V2H@<4> @I@Check@I@ the solutions for this@D6H@equation for truth value in the@D6H@orI@"24)"@22H@IF "TP$"@ID14H@THEN A=B@8F@AND A@G@=@R@B@D14H@SOMETIMES@6F@SOMETIMES":600:310jPP11200ZV136:V2156:502:"@5V2H@To solve a radical equation:@2D2H@<1> @I@Isolate@I@ the radical on one side@D6H@of the equation.":600:"@10V2H@<2> @I@hich A=B it is also true that@D2H@"TP$".@I16V2H@ IF A=B @D2HI@THEN "TP$"@D4H@ALWAYS":600:"@10V2H@HOWEVER, when "TP$" it is not@D2H@necessarily true that A=B. We need"\"@2H@to check each solution in the@D2H@original equation for truth value.@3D14Hs@D2H@NOT a root of the original equation.":600:310V136:V2116:502:V1124:V2156:H19:H288:1:500:H194:H2270:5:500:3:"@5V2H@When we solve radical equations we@D2H@use the principle that for all cases"TP$"A@G@#@R@=B@G@#@R@":"@2H@in wV2H@But observe this:@D2H@"RD$"@DB@X = -2":600"@14V3H@X = 4 (squaring both sides again.)":600:I02:"@"13I"V7H@ @DBI@4@I@":I51300:I5,I:"@14V7H@4@2D2H@But@U@"RD$"@DB@4 = -2 is NOT true (a principal"5"@2H@root is always positive), so 4 i0 the equation is true.@D2H@For all other values of X it is@D2H@false.":600:340:350/"@10V2H@This equation also has one solution:@D17H@"RD$"@DB@X = 2":600:"@14V2H@We square both sides to eliminate@D2H@the radical:@2D18H@X = 4":600:340:350:"@10 is not@D2H@under a radical sign, then solving@D2H@the new equation. However not all@D2H@solutions found this way are valid.":600:600:340:VT10:HT2:BL36:LN9:350V"@11V2H@This equation has one solution:@2D15H@25X = 1000":600:"@15V2H@When X = 4which a variable""@10V2H@An equation in which a variable@D2H@appears under a radical sign is@D2H@called a radical equation. Finding@D2H@solutions for a radical equation is@D2H@done by rewriting the equation in a""@2H@form in which the variable400:4103*M5100,5200,5300,5400@P5110,5120,1200V136:V268:502:"@5V2H@An equation's roots, or solutions,@D2H@are the values which its variables@D2H@can take to make the equation true.":600:V176:V2156:500:"@10V2H@An equation in 46000:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@11V10H@<3> EXAMPLE@10H13V@<4> SAMPLE PROBLEM@10H16V@<0> RETURN TO CONTENTS""@I10H22V@WHICH (0-4) ?? @I@":MN0:MX4:300:Kİ412:26:(4)"RUNAM5.2"MK:P1:36320M,(36320M)1:I$"@I@":W$I$"WRONG"I$", try again.":C$I$"CORRECT"I$R$(12):RD$"@L@"(16)"@R@"R$:X$"express":F$"fraction":RC$"radica":TT$(10)(10)(10):CF$"coefficient":D$"denominator":CJ$"conjugate":RA$"rational"410M0:P0:400:CqK141630:K136PSĹ511PS,32:PSPS1wrKK128:K47K58K45ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX630t622v(PS0)620:IN$"":I0PS:IN$IN$((512I))::IN(IN$):IN$(IN):"@R@":;D(X)((1)X)1:Y(X)D(8)1:35339:t your choice.":<XI11500::K(16384):MK155321:l35400,L(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32::16368,0nPSMXĺ"@"HTPSL"HI@"((512PS))I$;p328:(PSMX)(K141)(K136)624:PSMXĺ"@"HTPSL"H@"((512PS));)z#          ͵9͵͵7 ͵9͵/"խŠ "ӳ ͵Ӡ͵5͵'͵830976$"@21V1HLI@"19)"@RI@":dH11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2:yH19:H2270:500V1124:V2156:502:"@16V2H@Use "(91)(2)","(1)"] keys to light the correct@D2H@choice, and the "(91)"RETURN] key to @D2H@selec15V2H@Use the number keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Use "(91)(2)"] to backspace and@D2H@"(91)"RETURN] when you finish each part.":V1116:V2156:502 "@2H1V@"C"@5H@"P(P10)"@2HD@"M:3: 31051:414@R0K"VT"V@";:I31LN:"@"HT"H@"BL)""::8 cTOGL INVQ eBL1:VV1:V2V1LN1 g"@"V"V"HT"HI@"T$(BL)I$:328:KYK:16368,0:"@"V"V"HT"H@"T$(BL):KY141ı iKY149VV1:BLBL1:VV2357 kKY136VV1:BLBL1:VV1VV2:BLLN l359 m"@RA 5": H16368,02 IK(16384):K128329:V JPP(K149)(P0)(K136):332a K:331m L63900 MK136412:K149410:1300 T"@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINUE@I@":16368,0 Y320:K160345:"@22V1HI@"36)"@I@":* ^"PP1:410' >K8PP1:412:P12006 ?400:1300S @ZH(1):328:(36251)ı A(P(K136K149)(M4))305:K155ı C"@40X40YN@";:328:K155(K205C)ı D16368,0:326 E:325 F63900 G412:K1551200:26:(4)"RUNALGEB1T3T2T370:1 ,320:KK176:KMNKMX300:; 1307F 2:306R 363900[ 4312 6"@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE@I@" 7320 8KK128:K27(36251)0323:K21K8311:K21424577:30000*C(24576):639004 1000G0:1002:::7ZZ((TP)):TP$"@U@"RD$"@B@":I51ZZ:TP$TP$R$::TP$TP$"@"(ZZ)"BD@"(TP):<TPD(4)1:TPTP(TP4)3::***GENERATE2,3,5,7 F60:T1TP:60:T2TP:60:T3TP:T1T2T          -2":1)K::VT11:I02:HT32II:54:R(I4)K::TP(S1$):OK1:I13:OKOK(R(I)R(I3))::R(1)((R(1))):R(3)((R(3))):OKOK(R(1)T2)(R(3)T3)(R(2)TP):>(222)253Ħ?::3::"++ERROR++";::(222):"AT LINE #"(218)(219)256:"IN AM5.29U=TPT1T1T4:4:T5$TP$:TPT2T2T5:4:T6$TP$:T5$T3$T6$T4$;:CT2ĺT3;T1$:149m=TPT3T3T6:4:TP$=AN(T7T7T41(T70)S1(T2T2T5)S2(T3T3T6)(P(24MN))):=TP(1)D(8):TP$" "(44TP)" ":>VT8:I02:HT32II:54:R(IVT5:30:VT10:HT2:BL36:LN5:30:0:21,120265,120:3:38:136<P1:38:PM4:NC0:99:36256,NC:P1:39:NC0:130:36257,NC:3:(4)"RUNAM5.2-3"<I19:T(I)Y(1)::OK0:I13:OKOK(T(I)T(I3))::(OK0)145:=T1;T1$T3$T2;T2$T4$T3;T1$:14$(I1)"@11V"(312I)"H@"T9$T$(I1):T$(I1)::LN3:31:T(1)BLĺ"@10V2H@"W$:29:"@10V2H@"17):31:TX0;T(1)BLĺ"@10V2HI@WRONG@I@,the correct answer is:@I@"T$(T(1))I$:142;"@10V2H@"C$;29:NCNCTX:PP1:PPMıA<HT28:BL10:LN4:T7$"@D"(9(T49))"B@"TZ$TE$TE$"@D6B@"(T5):T$(T(2))"@U10B@"(T4)S1$T7$"@D"(9(T49))"B@"TZ$TE$TE$"@D6B@"(T5)o;T$(T(3))"@U8B@"(T1T2)S$(D(2)1)"@U@"RD$R$"@D2B@"(T6)"@D"(5(T69))"B@"TZ$"@D3B@"(T2T2T3):I02:T$S$((S1)):S2$S$(S1):"@5V32H@"T1"@D3B@"TZ$"@2D5B@"T2;S1$"@U@"RD$"@DB@"T39I13:T(I)D(3)::T(1)T(2)T(1)T(3)T(2)T(3)137:((T5T1)(T5T1))T4T2:T5T5T1:T11:T7$"@12V@("(T1)"*""@U@"RD$"@DB@"(T3)")":T$(T(1))"@U10B@"(T4)S2$B@ @RB@ ":39:PP1:38:44:V2V1:V136:42:V276:42:H2192:42:S$(0)"+":S$(1)"-"K96:"@I5V2H@Which "F$" below was @D2H@written by "RA$"izing @D2H@the "D$" at the @D2H@right using "CJ$"s? @I@":TX1:T4T1T2:T5T2T2T3:T6T1T3:S1D(2)1:S124H@again.":29:"@5V24HL@"5)"@R8V32H@ @2F@ @3D5B@ @2F@ ":1517TP432(TP43):OKĺ"@I5V24H@WRONG@I@, the@D24H@answer is:@2D2B@"T2;(TP)"@2F@"T3"@3D5B@"T2;(TP)"@2F@"T37OKĺ"@5V24H@"C$Y8NCNCTX:29:(P(22(M4)))131:T9$"@11VL@ @D5$"."I$:P06PP1:HT24:VT5:LN7:BL14:30:38:T1D(8)1:T2D(8)1:5:T3TP:150:S1TP:S1$(TP$,2,1):TX1:"@8V26H@"T1"@D24H@"TT$TE$"@2D5B@"T2;S1$"@U@"RD$"@DB@"T3"@9V30H@* "TT$TE$"@7V34H@"RD$"@10V34H@"RD$:151A7OKTX0:"@5V24HI@WRONG@I@, try@D123539:61&5PM9:130:39:61 637:V2V1:V136:42:H2165:42:I$:HT2:VT5:BL21:LN9:30:"@6V2H@For the example at@D2H@the right, fill in@D2H@the "CJ$" of the@D2H@"D$" to create@D2H@a multiplier which":"@2H@will "RA$"ize the@D2H@original "D@D16H@"5):46::"@16H@"TT$TE$"@2D16H@"T2$"@10V14H@*":29{4~T4T1T2:T5T2T2T3:((T5T1)(T5T1))T4T2:T5T5T1:T115"@17V16H@This eliminates all@D2H@"RC$"ls from the "D$".@10V23H@= "TT$TT$TT$"@U9B@"T4;S2$"("T1"*"T3$")@2D29H@"T5:29:P4PP1:B@"T1$:29:"@14V2H@We can multiply the original@D2H@"F$" by 1 in the form of the:4}"@2H@original "D$"'s "CJ$"@D2H@over itself.":I07:"@12V"8I"H@"T1$:46::"@12V7H@"T1$:I110:I$"@13V17H@^":46::"@12V16H@"T2$"@D17H@ ":I03:"@16H"12I"V@"T2$"42:"@5V2H@Rationalize the "D$" of this@D2H@"F$":":38:S$(0)"+":S$(1)"-"2{38:6:VT8:HT2:BL36:LN5:30:TP((P2)(P2)):S1$S$(TP):S2$S$((TP))Q3|T3$"@U@"RD$"@DB@"(T3):T1$" "(T2)S1$T3$:T2$(T2)S2$T3$:"@9V10H@"T1"@D3B@"TT$TE$"@2D66V2H@To "RA$"ize an ir"RA$" "RC$"l@2D2H@"D$" of the form X @G@+@R@ @U@"RD$"@DB@Y,@2D2H@multiply the "F$" by 1@2D2H@represented as a "F$" whose"1y"@D2H@numerator and "D$" are the@2D2H@"CJ$" of the first binomial.":11K2zP461:V136:V2156:43:V260:ultiply by the@D21H@"D$"'s@D21H@"CJ$" over@D21H@itself (=1)....@4D12H@*"I$T2$I$:29:T2$I$T3$I$:43:"@16V2H@...and the result is a "F$" with@D2H@a "RA$"ized "D$".";:"@31H@This@D2H@new "F$" is in simpler form.@6U23H@=":111xP161:V136:V2140:43:"@4)S2$"("(T3)"*@U@"RD$"@DB@"(T2)")""@D26H@"TZ$TE$TE$"@2D29H@"(T1)"@G@2@R@-"(T2)/vH19:H2144:V136:V276:42:"@5V2H@Here is a "F$"@D2H@with a square root@D2H@binomial as its@D2H@"D$":"I$T1$I$:29:T1$:43:V1124:V21560w"@5V21H@M @D3B@ @R11V6H@"(T3)"@D4H@"TZ$"@2D5H@"(T1)S1$"@U@"RD$"@DB@"(T2):T2$"@L10V15H@ @13V2B@ @R11V15H@"(T1)S2$"@U@"RD$"@DB@"(T2)"@D5B@"TT$TE$"@2D5B@"(T1)S2$"@U@"RD$"@DB@"(T2)b/uT3$"@10V26HL@ @13V26H@ @R11V26H@"(T factors differ from each@D2H@other only in the sign of their@D2H@operators and are called @I@"CJ$"s@ID2H@of each other. The product of the@D2H@"CJ$"s is a "RA$" number.":11-s6:T4T1T3:150:S1$(TP$,2,1):S2$"+":S1$"+"S2$"-".tT1$"@11V4HL@ putations.":11,qV136:V2156:43:"@5V2H@The general equation:@2D8H@(A-B) * (A+B) = A@G@2@R@-B@G@2@R2D2H@applies to binomials containing@D2H@square root components as:@2D6H@(X-@U@"RD$"@DB@Y) * (X+@U@"RD$"@DB@Y) = X@G@2@R@-Y.":45-r"@14V2H@These twoare RATIONAL numbers)@D2H@and would prefer being able to work",p"@2H@with a "RA$" "X$"ion instead.@D2H@This is often true when the "RC$"l@D2H@"X$"ion is the "D$" of a@D2H@"F$". Such a "F$" is not in@D2H@simplest form and is difficult to@D2H@use in com)R(4)1(R(3)0))S2(R(5)R(5)R(6)1(R(5)0)))AN):R*mM110,120,122,129h*nP111,113,115,61>+oV136:V2156:43:"@6V2H@At times you may encounter a "RC$"l@D2H@"X$"ion of the form:@2D13H@X+@U@"RD$"@DB@Y or X-@U@"RD$"@DB@Y@2D2H@(where X and Y PPMPP1:HT2:VT8:BL36:LN6:30:38:1002)j)kR(5)0:R(6)0:HT2:I3142(CT2):47:R(I3)IN:((I32)(I32))ĺ"@"VT"V"HT"H@ @"(IN$)"B@"IN;*lHTHT6((I32)(I32))::OK(R(2)0R(4)0R(6)0)((R(1)R(1)R(2)1(R(1)0)S1(R(3)R(3U@"RD$R$R$x(f107:TX1:OKTX0:"@12V2H@"W$:29:"@12V2H@"17)"@U2H@ @3F@ @2F@ @3F@ @2F@ @3F@ ":107(gOKĺ"@12V2HI@WRONG@I@,@D2H@the answer is: "T7;T1$T3$T2;T2$;:CT2ĺT4$T3"*@U@"RD$"@DB@"T6(hOKĺ"@12V2H@"C$,)iNCNCTX:29:S2TP:T4$TP$:T7T1(CT3)S2T3:T1T3T7T2(CT2(T7T3T2T3))100:T1$"*@U@"RD$"@DB@":T2$T1$(T5):T1$T1$(T4):"@9V2H@"; (eHT2:VT11:150:S1TP:T3$TP$:CT146,147,147,147:"@11V6H@*@U@"RD$R$R$"@D@"T3$"@4F@*@U@"RD$R$R$;:CT2ĺ"@D@"T4$"@4F@*@6"*"T6$")":11+&bP960:PM9:99:39:61&c36:V136:42:V160:42:"@5V2HI@Combine the "RC$"l "X$"ions in @D2H@simplest form as indicated:"9)I$:MX3:L1:MN(M4)'dCT((P1)(12MN)1):6:T4T1:T5T2:T6T3:T1D(4)1:T2D(4)1:T3D(4)1:150:PT3:4:TP$" + ";:TPT4:4:TP$:45:TPT(5):4:T5$TP$:TPT(6):4:T6$TP$:"@11V3H@("T(1)"*"T5$")+("T(2)"*"T6$")-("T(3)"*"T6$")+("T(4)"*"T5$")":45&a"@14V5H@"(91)"("T(1)"+"T(4)")*"T5$"] + "(91)"("T(2)"-"T(3)")*"T6$"]":45:"@17V9H@("T5"*"T5$") + ("T$_I16:5:T(I)TP::T5T(1)T(4):T6T(2)T(3):T(1)T(4)T(2)T(3)T(5)T(6)T5T695:T1T(1)T(1)T(5):T2T(2)T(2)T(6):T3T(3)T(3)T(6):T4T(4)T(4)T(5)%`V136:V2156:43:"@5V2H@Add or "ST$":":TPT1:4:"@8V4H@"TP$" + ";:TPT2:4:TP$" - ";:T@"T1$" * "T2"@2D10H@=@6F@"TP$" * "T2"@D12B@"TZ$TZ$"@14V2H@<1> Simplify each "RC$"l.":45$^"@2H@<2> Combine like "RC$"ls.":45:"@12V11H@("T1$"-"TP$") * "T2"@UF@(Distributive@D26H@Property)@4D2H@<3> Show the sum or difference of@D6H@unlike "RC$"ls.":11"@U23H@(Distributive@D24H@Property)@3D2H@<2> Combine like "RC$"ls.":45:11"\6:T4T2T2T1:T5T2T2T3:V136:V2156:H19:H2270:42:TPT4:4:"@5V2H@Add or "ST$":@2D4H@"TP$X#]TPT5:4:"@D3H@-"TP$"@D3H@"TZ$:45:TPT1:4:T1$TP$:TPT3:4:"@7V10H@=@6F156:43:"@5V2H@Add or subtract:":TPT2T2T3:4:"@7V4H@"TP$"@2D4H@+"T1"*@U@"RD$"@DB@"T3"@D4H@"TZ$:45:TPT3:4:"@7V12H@= "T2" * "TP$"@2D12H@=@15H@"T1" * "TP$"@D14H@"TT$TZ$"@14V2H@<1> Simplify each "RC$"l.":45K"["@12V"15((T1T2)9)"H@"T1T2" * "TP$ "TP$"@D16H@"TT$TZ$:45 Y"@12V"17(T39)"H@"T3" * "TP$:42:45:TPT7:4:"@17V16H@Subtraction works in@D2H@the same way.":45:"@7V29H@"T1" * "TP$"@2D28H@-"T2" * "TP$"@D28H@"TT$TZ$:45:"@D"29(T40)"H@"T4" * "TP$:H1192:H2214:42:11!Z6:V136:V2H@+"T2" * "TP$"@D4H@"TZ$TT$:45:"@D"5(T39)"H@"T3" * "TP$:5:42:45 X"@14V2H@"T1" and "T2" are "CF$"s of the@D2H@"RC$"l. We add "CF$"s of@D2H@like "RC$"ls to get the "CF$"@D2H@for the sum.":H1108:H2129:TPT6:4:"@7V17H@"T1" * "TP$"@2D16H@+"T2" *7D(98)1:T5T6T5T7T6T785:T10:I29:T2II:T1(T5T2T6T2T7T2T1)::T185VT1Y(1):T2Y(1):T1T2T1(T2)86:H19:H2270:V136:V2156:42:V152:V2108:H124:H245FWT3T1T2:T4T1T2:TPT5:4:"@5V2H@Add or "ST$":@2D5H@"T1" * "TP$"@2D4C$"ls@D2H@(different index and/or "RC$"nd):@2D2H@<1> Simplify each "RC$"l.":45:"@D2H@<2> Combine like "RC$"ls (if any).":45:"@D2H@<3> Indicate the sum or difference@D6H@of the unlike "RC$"ls.":45:11TP85,90,92,95,61VUT5D(98)1:T6D(98)1:T@To add or "ST$" ":V152:V2132:43:(P1)83:T$"like "RC$"ls@D2H@(same index and same "RC$"nd):@2D2H@<1> Add or "ST$" the "CF$"s@D6H@of the "RC$"ls.":45R"@13V2H@<2> Multiply the resulting@D6H@"CF$" by the common@D6H@"RC$"l.":11ST$"unlike "R"@15V2H@"T1"*@U@"RD$"@DB@"T3"+@U@"RD$R$R$"@D3B@"T4;:45:"@14H@= "T1"*@U@"RD$"@DB@"T3" + "T2"*@U@"RD$"@DB@"T3"@2D2H@The simplified "X$"ions can then@D2H@be added or "ST$"ed as usual.@3U@";:45:"@30H@= "T5"*@U@"RD$"@DB@"T3:45:11PP261QT$"@7V2Hn or@D2H@"ST$"ion of their "CF$"s.":45:"@D2H@Many times we wish to combine"N"@2H@"RC$"l "X$"ions which appear to@D2H@be unlike, but which actually can be@D2H@simplified and shown to be like.":45:T1Y(1):5:T2TP:5:T3TP:T4T2T2T3:T5T1T2O1:VT10:47:"@10V21H@ @"(IN$)"B@"IN:I1IN:HT31:47:"@10V31H@"7)"@U2B@ @D7B@"(IN)T1$"Y":OK(INT(7PM))(I1(T(1PM)T(4PM)(2T(4PM)(PM1)))):MMV136:V2156:43:"@5V2H@We have just seen that like "RC$"ls@D2H@can be combined by the additio"W$:29:"@12V2H@"17)"@9V31H@"7):76IOKT2$(T(1PM)T(4PM)(2T(4PM)(PM1)))T1$"X"S$(PM2)(T(7PM))T1$"Y.":"@12V2HI@WRONG@I@, the answer is "T2$JOKĺ"@12V2H@"C$KPM229::11L"@21H10V@ "T1$"X "S$(PM2)" "T1$"Y":HT2)"X":T$(1)"Y":S$(0)"+":S$(1)"-"GMX3:L1:PM02:T1T(13PM)T(23PM):T2T(13PM)T(23PM):HT2:VT8:BL36:LN5:30:"@9V2H@"T(PM1)T1$"X "S$(PM2)" "T(43(PM0)PM)T1$T$(PM0)" "S$(PM1)" "T(43(PM0)PM)T1$T$(PM0)" ="&H76:OKĺ"@12V2H@(4)"@3D3H@"T(2)"@15H@"T(5)"@21H@"T(2)T(5)"@3D3H@"T(3)"@15H@"T(6)"@21H@"T(3)T(6)I$:45:11#F36:V136:42:"@5V2H@The same procedure allows us to add@D2H@and "ST$" "RC$"l "X$"ions --@D2H@treat each "RC$"l as a variable:":145:T1$"*@U@"RD$"@DB@":T$(0Y":45D"@12V3H@"T(2)"*X + "T(8)"*Y - "T(5)"*X = "T(2)T(5)"*X + "T(8)"*Y":45:"@15V3H@"T(3)"*X - "T(9)"*Y + "T(6)"*X = "T(3)T(6)"*X - "T(9)"*Y":45[E"@17V2H@We add or "ST$" the "CF$"s@D2H@of the like variables.@I9V3H@"T(1)"@9H@"T(4)"@21H@"T(1)TM,(36320M)1:38:39(@3:C,65,109;AM66,80,84,98NBP67,70,77,61CV136:V2156:43:"@5V2H@We already know how to add or@D2H@subtract "X$"ions containing@D2H@variables:":145:"@9V3H@"T(1)"*X + "T(4)"*X + "T(7)"*Y = "T(1)T(4)"*X + "T(7)"*M0:P0:38:C5144:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@11V10H@<3> EXAMPLE@10H13V@<4> SAMPLE PROBLEM@10H16V@<0> RETURN TO CONTENTS@I10H22V@WHICH (0-4) ?? "I$>16:KK176:K0K462:Kİ40:3:(4)"RUNAM5.2"?MK:P1:36320RONG"I$", try again.":C$I$"CORRECT"I$:TE$(10)(10):ST$"subtract";R$(12):RD$"@L@"(16)"@R@"R$:X$"express":F$"fraction":RC$"radica":TT$(10)TE$:CF$"coefficient":D$"denominator":CJ$"conjugate":RA$"rational":TZ$TT$TT$<39=8:((K43K45)HT33)(K47)(K58)(HT33)ĺ"@BI@"(K)I$;:KYK:558HT33(K43K45K13)ĺ"@6V24HI@USE '+' OR '-'"I$:45:"@6V24H@"14)"@"VT"V4B@";:559K1354:KKY:"@B@"(K):I:D(X)((1)X)1:Y(X)D(8)1:35339:I$"@I@":W$I$"W;,2K14153:K136PSĹ511PS,32:PSPS1y3KK128:K47K58K45ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX534485(PS0)47:IN$"":I0PS:IN$IN$((512I))::IN(IN$):IN$(IN):"@R@":6"@"HT"H"VT"VI@ "I$;:KY48E723:KK12t your choice.":<-I611500::K(16384):MK15517:L.Z199::/35400,L(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32:0PSMXĺ"@"HTPSL"HI@"((512PS))I$;123:(PSMX)(K141)(K136)49:PSMXĺ"@"HTPSL"H@"((512PS))(30976&)"@21V1HLI@"19)"@RI@":f*H11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2:z+H19:H2270:42,V1124:V2156:43:"@16V2H@Use "(91)(2)","(1)"] keys to light the correct@D2H@choice, and the "(91)"RETURN] key to @D2H@selech each part.":V1116:V2156:43 %"@15V2H@Use the number keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Press "(91)"RETURN] when you@D2H@finish each part.":V1116:V2156:43 &"@2H1V@"C"@5H@"P"@2HD@"M:3: '31051:41 # !KY149VV1:BLBL1:VV231N "KY136VV1:BLBL1:VV1VV2:BLLNV #32 $"@15V2H@Use the number keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Use "(91)(2)"] to backspace and@D2H@"(91)"RETURN] when you finis40:K14939:64h "@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINUE"I$:16:K16029:"@22V1HI@"36)I$: "@R0K"VT"V@";:I31LN:"@"HT"H@"BL)"":: BL1:VV1:V2V1LN1 "@"V"V"HT"HI@"T$(BL)I$:23:KYK:16368,0:"@"V"V"HT"H@"T$(BL):KY141ıK149)(M4))7:K155ıB "@40X40YN@";:23:K155K205ıU 16368,0:21_ :20i 152 40:K15561:3:(4)"RUNALGEBRA 5" 16368,0 K(16384):K12824: PP(K149)(P0)(K136):27 :26 152 K136152 13q "@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE"I$y 16 KK128:K27(36251)18:K21K812:K21PP1:39 K8PP1:40:P61 38:64 23:(36251)ı (P(K1367$24577:30000:C(24576):152,58;0:1002:ZZ((TP)):TP$"@U@"RD$"@B@":I51ZZ:TP$TP$R$::TP$TP$"@"(ZZ)"BD@"(TP):TPD(4)1:TPTP(TP4)3:5:T1TP:5:T2TP:5:T3TP:T1T2T1T3T2T36:9:8                     IN AM5.2":"@4B@2@DF@"T8$=(==@35339::I$"@"(45)"H@HI":I$:=PÃSIMPLIFYING RADICALS,ADDITION AND SUBTRACTION,CONJUGATES,RADICAL EQUATIONS,RADICAL EXPRESSIONS TEST=(222)253Ħ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is:@2D2F@"T1" * @U@"RD$"@3B@"T7"@FD@"T2:9(***C1M4P4-6 ALG9(T(1)D(4):I24:(T(I)D(4):OK0:J1I1:OKOK(T(I)T(J))::(OK0)10417::S3TT410402:400:T7D(2)1:T5(T1T7T2):T6(T3T7T4):OK0:8("@8V2H@"RD$R$R$R$"@6B@"T7"@DF@"T5"@D2H@"T1$T1$T1$"@D2H@"RD$R$R$R$"@6B@"T7"@DF@"T6"@2U8H@="T1$T1$T1$T1$T1$"@U12H@* @U@"RD$R$R$"@5B@"T7"@4D12H@* @U@"RD$R$R$"@5B@"T7:C9(620:"@"HT"H"VT"V@ @"RD$R$R$"@5B@2@DF@"T2"*"T4"@2D5B@"T3"*"T4::***IVQ7(TPD(3)1:TPTP(TP4):c7(***C1M4 ALGS7(***10401-10404,P1-3///10405-10409,DISPLAY I-O///10410,'WRONG' MESSAGE AFTER LAST TRY=8(10115:T1TP:10115:T2TP:10115:T3TP:10115:T4TP:T1T3T2**C1M3 ALGSx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hich of the following "T$(T4T502:HT158I:350::2432+1 (M4)410:r1p36251,2:P1:PM1:2410:36255,NC:24576,5:26:(4)"RUNAM5.2-2":1'***C1 ROUTINES1t' ***C1M1P1,2 ALG'2~'10115:T1TP:10115:T2TP:10115:T3TP:T1T2T1T3T2T310110:510:V136:V2124:500:V260:T$BL36:350:"@17V14H@"W$:340:510:355:T(1)BLBL36:350:"@I16V17H@WRONG@I@,@D3H@see correct answer lighted above."I$T$(T(1))I$0 T(1)BLBL36:350:"@17V17H@"C$1 340:PP1:NCNCTX:P((3PM)1(M4))İ510:HT2:VT10:BL12:LN5:350:BL7:I00:10402:(T2T3)(T1T4)2432:T5T7T1:T6T7T3:T8T2T4:T9T3T4:T0T6T4:"@11V"4(T510)"H@"T5" * @U@"RD$R$"@4B@2@DF@"T2/ "@4H@"TT$TT$TT$"@2D"4(T610)"H@"T6" * @U@"RD$R$"@4B@2@DF@"T4:10420:I13:T$(I)::LN3:355:HT2:VT160 T(1)BLTX0:0:V276:500:H2102:"@5V2H@This "RC$"l@D2H@"X$"ion@D2H@is not fully@D2H@simplified:":500:"@I5V15H@Which of these is the @D15H@simplified form of the ". "@15H@"RC$"l "X$"ion at @D15H@the left?"14)I$:V176:V2124:500:H1157:H2214:500}/ TX1:4350:"@17H16VI@WRONG@I@,@D3H@see correct answer lighted above":I$T$(OK)I$-| BLOKHT2:VT16:BL36:LN3:350:"@17V13H@"C$:NCNCTX-} 340:PP1:P((2PM)1)HT2:VT10:BL12:LN5:350:BL5:I03:HT156I:350::2422.~ 400:410:510:V136:50H1144:H2186:500:H2227:500,v 510:10402:"@14V5H@"T3" *@2U3B@"TT$(10)"@U4B@"T1" *":S1T2:S2T4:S3T7:S4T7:55:"@10V9H@"TP$,x 10415:LN4:TX1:355:BLOKTX0:HT2:VT16:BL36:LN3:350:"@17V13H@"W$:340:LN4:510:355K-z BLOKBL36:LN3:+t 400:410:510:V136:500:V276:500:H2102:"@5V2H@This "RC$"l@D2H@"X$"ion@D2H@is not fully@D2H@simplified:":500:"@I5V15H@Which of these would be@D15H@useful in simplifying @D15H@the ";,u RC$"l "X$"ion @D15H@at the left?"11)I$:V176:V2124:500:(TPT3)(INT4)ĺ"@9V21HI@WRONG@I@,@D21H@the@D21H@answer@D21H@is:@2U4F@"T1$T1$T1$T1$"@U7B@"T1" * @U@"RD$"@3B@"T7"@FD@"T2"@3D29H@"T3" * @U@"RD$"@3B@"T7"@DF@"T4*r TPT3INT4ĺ"@12V21H@"C$*s 340:PP1:NCTX:P(PM1)HT2:VT7:BL35:LN7:350:2412$R$R$:HT9:10405H)n (TPT1)(INT2)10410:TPT1INT2ĺ"@9V21H@"C$)o 340:"@9V9H@ "T1"@16H@"T2" ":I02:"@L21H"82I"V@"8):)p HT9:VT12:10405:(TPT3)(INT4)TX0:T2$:340:"@0C@"T2$"@15C12V9H@ @4F@ @U3B@"R$R$R$:HT9:10405*q 2410:1200(j 365:V136:500:"@5V2HI@Simplify the "N$" and"10)"@D2H@"D$" of this "X$"ion: "I$:T1$(10)(10):T2$"@10V21H@"W$)l 10402:10404:L1:MX3:HT9:VT9:10405:TX1:(TPT1)(INT2)TX0:T2$:340:"@0C@"T2$"@15C9V9H@ @4F@ @U3B@"R "TT$"@2U3B@"RD$"@3B@2@DF@"T4"@2D3B@"RD$"@3B@2@DF@"T4"@27H@";:10384:340:"@18V2H@<4> Simplify the resulting "F$".":HT9:VT5' BL29:LN5:350:"@9H@";:10384:"@7V22H@= "TT$TT$T1$"@U7B@"T7"*@U@"RD$R$"@4B@2@DF@"TM"@2D28H@"T8:310 (` P91100:PM3:10382:600:340:HT9:BL29:LN5:VT5:350& HT2:BL36:VT11:LN8:350:"@11V2H@<3> If the "D$" still@D6H@contains a "RC$"l, multiply@D6H@the whole "F$" by 1 in a@D6H@form which will eliminate the@D6H@"RC$"l in the "D$"."' "@2D9H@";:10382:"@7V19H@*252:500% "@12V2H@<1> Change the form of the "F$"@D6H@under the "RC$"l to a "RC$"l@D6H@divided by a "RC$"l.@7V9H@= "TT$T1$T1$"@5B2U@"RD$R$R$"@5B@2@DF@"T5"@8V12H@"RD$R$R$"@5B@2@DF@"T6:600)& "@16V2H@<2> Simplify both the "N$" and@D6H@"D$".@19H@";:24H@2@5F@2@2F@2"$ 600:"@13V26H@"T1" * @UL@"RD$"@3B@2@DF@"T2" * "T3:600:"@15V32HL@"RD$"@3B@2@D6B@"T4" *@34H@"T2:600:310 % T1$(10):30:"@5V2H@Simplify@D2H@this@D2H@"RC$"l@D2H@"X$"ion: ";:10380:340:HT2:VT5:BL16:LN4:350:"@8V3H@";:10380:H:HT24:BL14:350# T4T1T3:"@9V2H@<3> Express the@D6H@result as the@D6H@product of the@D6H@nth root(s) of@D6H@the isolated@D6H@"FC$"s and the@D6H@remaining "RC$"l@D6H@"X$"ion."$ "@8V24HL@"RD$R$R$" "RD$RD$R$R$"@D26H@"T1"*"T1"@3F@"T2"@2F@"T3"*"T3"@U2> Let n= the index.@D6H@If any "FC$"@D6H@appears n or more"# "@6H@times, isolate@D6H@the "FC$" in@D6H@groups of n under@D6H@another "RC$"l.@2U24H@"RD$R$R$" "RD$RD$R$R$"@24H@2@5F@2@2F@2@D26H@"T1"*"T1"@3F@"T2"@2F@"T3"*"T3:340:HT2:VT8:LN11:BL21:350V2156:500:H2165:500:H2270:V260:500:"@5V2H@Simplify this radical@D2H@"X$"ion:@U29HL@"RD$T1$"@29H@2@DF@"T4<" 600:"@8V2H@<1> Find the prime@4FL@"RD$"@3B@2@D6H@"FC$"s under the@D6H@"RC$"l.@U26H@"T1"*"T1"*"T2"*"T3"*"T3:181,70244,70:600:"@12V2H@ No "RC$"ls appear in the@D6H@"D$" of the "X$"ion.":600:310t P2310,2310,2320,2320,1200  10115:T1TP:10115:T2TP:10115:T3TP:T1T2T1T3T2T32310:T1$R$R$R$:T4999T1$T1$R$p! T4T1T1T2T3T3:H19:H2270:V136:200V144:V2140:H19:H2270:500:"@6V2H@A "RC$"l "X$"ion is in simplest@D2H@"RC$"l form when:@2D2H@<1> No "FC$" of the "RC$"nd is an@D6H@integer raised to the nth power@D6H@(n is the index of the "RC$"l).":600T "@13V2H@<2> The "RC$"nd is not a "he "RC$"l so that no@D6H@portion of the "D$" is@D6H@"X$"ed in "RC$"l form."T$T$(1)T$(2):LN2:355:BL2ĺ"@14V2H@"C$[BL1ĺ"@16V2HI@WRONG@I@, the second is in simpler form@D2H@because its "D$" is not @D2H@"X$"ed in "RC$"l form."\310P2210,1he@D6H@"RC$"l sign) is not a "F$"."T$T$(1)T$(2)VWLN2:355:BL1ĺ"@14V2H@"C$:2138X"@16V2HI@WRONG@I@, the first is in simpler form@D2H@because it contains no "F$"al@D2H@"RC$"nds."9)tZ340:VT8:HT2:BL36:LN11:350:10140:"@8V2H@<3> Rewrite t in simpler form @D2H@because the "FC$" "T1" has been moved@D2H@out from under the "RC$"l sign. @DL2H@"16)"@R@"/V340:10135:VT8:BL36:LN8:HT2:11,124268,124:512:350:3:"@8V2H@<2> Rewrite the "RC$"l so that the@D6H@"RC$"nd (the number under tt rules to@D2H@simplify "RC$"l "X$"ions:@2D2H@<1> Remove as many "FC$"s of a@D6H@number";S" as possible from under@D6H@the "RC$"l sign."T$T$(1)T$(2):LN2:355:0:BL2ĺ"@14V2H@"C$:2134sT11,124268,124:"@14V2HI@WRONG@I@, "T1"*@U@"RD$"@DB@"T2" is0V19H@"17):355-NBL4ĺ"@10V19H@"C$:310P"@10V19HI@WRONG@I@, "T4T5" is the@D19H@simplest because it@D19H@is written as a@D19H@single integer.":310YRT$"@12V2H@Which is simpler?:@26H@or":510:V136:500:10130:"@5V2H@We use several consisten ":10100:BL4ĺ"@10V19H@"W$:340:"@10V19H@"17):355UDBL4ĺ"@10V19H@"C$:310F"@10V19HI@WRONG@I@, ALL OF THESE@D19H@"X$"ions mean@D19H@the same as "T4T5".":310HT$"is the@D2H@simplest way to "X$" ":10100:BL4ĺ"@10V19H@"W$:340:"@10H16V@<0> RETURN TO CONTENTS"^"@I10H22V@WHICH (0-4) ?? @I@":MN0:MX4:300:Kİ412:1100MK:P1:36320M,(36320M)1:400:4103:C2000,,,,6000M2100,2200,2300,24004P2110,2120,2130,12008>T$"means the@D2H@same as the number24(T$)2:"@2VI@"T$I$:Cİ412:26:(4)"RUNALGEBRA 5":rgC1C51130:24576,C:26:(4)"RUNAM5.2-"2(C4)j410:C51300M0:P0:400:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@11V10H@<3> EXAMPLE@10H13V@<4> SAMPLE PROBLEM@12,14249,14842,15420,154:"@18V3H@<0>"y[59,3259,159:60,3260,159:"@20H5V@CONTENTS@6V@":I15:"@9H@<"I"> "C$(I)`:I:"@9H18V@<0> RETURN TO ALGEBRA MENU":"@2V7HI@"31)"@11H21V@WHICH ONE (0-5) ?? "I$cMN0:MX5:300:CK:400:T$C$(K);e:"numerator":X$"express":FC$"factor":RC$"radica":TT$(10)(10)(10)LP0:M0:C0:400:H117:H245:I04:V14516I:V2V112:500:"@3H"62I"V@<"I1">":V14516(I1)16(I4)(V32,V232,V136,V13:31,V231,V127,V13::20,15413,14820,1424I0PS:IN$IN$((512I))::IN(IN$):IN$(IN):"@R@":D(X)((1)X)1:35339:C$(5):I15:C$(I)::I$"@I@":W$I$"WRONG"I$", try again.":C$I$"CORRECT"I$HR$(12):RD$"@L@"(16)"@R@"R$:TB$"@D2B@":F$"fraction":D$"denominator":N$PSL"HI@"((512PS))I$;ep328:(PSMX)(K141)(K136)624:PSMXĺ"@"HTPSL"H@"((512PS));qK141630:K136PSĹ511PS,32:PSPS1rKK128:K47K58K45ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX630t622:v(PS0)620:IN$"":URN] key to @D2H@select your choice.":RXI11500::K(16384):MK155321:ubTPD(4)5:TPTP(TP10)118:g2(TP10)(TP15)3(TP138)4(TP133):l35400,L(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32::16368,0nPSMXĺ"@"HT"M:3:31051:414"30976?"@21V1HLI@"19)"@RI@":H11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2:H19:H2270:V1124:V2156:500'"@16V2H@Use "(91)(2)","(1)"] keys to light the correct@D2H@choice, and the "(91)"RET359m"@15V2H@Use the number keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Use "(91)(2)"] to backspace and@D2H@"(91)"RETURN] when you finish each part."oH19:H2270:V1116:V2156:500"@2H1V@"C"@5H@"P(P10)"@2HD@I2051:I:2:I$38)I$:I:2:38)::Cc***TOGGLE INVERSE***\eBL1:VV1:V2V1LN1g"@"V"V"HT"HI@"T$(BL)I$:328:KYK:16368,0:"@"V"V"HT"H@"T$(BL):KY141ıiKY149VV1:BLBL1:VV2357kKY136VV1:BLBL1:VV1VV2:BLLNl):332K:331L63900>MK136412:K149410:1300~T"@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINUE@I@":16368,0Y320:K160345:"@22V1HI@"36)"@I@":^"@R0K"VT"V@";:I31LN:"@"HT"H@"BL)""::`VT15:HT1:BL38:LN5:350)b)ı/ A(P(K136K149)(M4))305:K155ı\ C"@40X40YN@";:328:K155(K205C)ıp D16368,0:326{ E:325 F63900 G412:K1551200:26:(4)"RUN ALGEBRA 5": H16368,0 IK(16384):K128329:JPP(K149)(P0)(K13600 4312o 6"@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE@I@"x 7320 8KK128:K27(36251)0323:K21K8311:K21PP1:410 >K8PP1:412:P1200 ?400:1300 @ZH(1):328:(362516 ARE SET UP W/IN ROUTINE) 7S1$(S1):S2$(S2):S3$(S3):S4$(S4):TP$RD$"@3B@"S3$"@DF@"S1$"@D3B@"TT$"@D3B@"RD$"@3B@"S4$"@DF@"S2$::***GIVENS1-4,SETS UP TP$(S1,2 MUST BE <10 ,320:KK176:KMNKMX300: 1307 2:306 3639((S2)):RD$"@B@";:I1S5:R$;::"@"2S5"B@"S3"@DF@"S1"@D"2S5"B@";:I1S52:(10);::"@D"2S5"B@"RD$"@B@"; 4I1S6:R$;::"@"2S6"B@"S4"@DF@"S2: 5***50-52 VBLS ARE S1-6...NUMERATOR,DENOMINATOR,NUM... ROOT,DEN... RT,LEN OF S1,LEN. OF S2(S5,SR(I)5):IIOK::TEPR(1):PR(1)PR(OK):PR(OK)TE:T7PR(1)O "T4PR(3):T2PR(2) $T9D(2)1:T33D(2):T3T3(T35):T8T3T4T9:T9T4T3T4(T8(T8))(T8T7(T8T7))32:T1T9T7:(T1T3)36 &TMT2T4:T5T1T1T2:T6T3T3T4:p 2S5((S1)):S6 624577:30000?PR(1)2:PR(2)3:PR(3)5:PR(4)7:63900I 1000\0:1002:::H19:H2270:V136:V2156:500:V284:500: ***C1M3P2 ALG9 I15:T1D(4):T2D(4):T3PR(T1):PR(T1)PR(T2):PR(T2)T3::T7PR(1):T73OK0:I24:OKOKI(P               .008723,0011.333333333,0011.645322222,0011.527272727,000072.4494897,000019.7071067,000042.8284271,0000.212212221,0000.050605070,0000.989989998-(222)253Ħ-:26::"++ ERROR ++";::" "(222):"AT LINE #"(218)(219)256:"IN AM5.1-2":o:400:Cİ410:26:(4)"RUNALGEBRA 5"?,26:(4)"RUNAM5.2-3"W,@***ROUTINES FOR C3,600:"@"VT"V@"T$TE$"?"T4$;:600:I((T3$,1)):TE$(I);:T3$(T3$,(T3$)1):Iİ600:T1$::M34316:4419,-PÃ10013,10015,100111,01012.803,010127.89235,0101@ = "T2T2" (too large)":IN$"IR"IN$+M"@15V8H@Therefore@UL@"(16)"@R15V@"TP" is "I$IN$I$".":131,118153,118:340:(PPM)PP1:"@6V34H@ @ID7B@ @I10V3H@"9)"@L5H12V@"16)"@14V5H@"16)"@R@":4422+N%,NC4:P1:LM2:100:PM4:4410:36254,NC:P0328:KK128:K82K734424:"@I7V30H@"(K)I$:J(TP(TETE)):I(J(K73))((J)(K82)):ITP$"@I@WRONG@I@":NCNC(M4)*JIN$"RATIONAL":"@10V3H@"TP$:Jĺ"@5H13V@"TE"@G@2@R@ = "TP:4429%+L"@5H12V@"T1"@G@2@R@ = "T1T1" (too small)@5HD@"T2"@G@2@R:3)D"@I5V2H@Press "(91)"R] if the "S$" @D2H@of this "N$" is "R$"; @D2H@"(91)"I] if it is "IR$":"6)I$:227,37227,67:228,37228,67)F400:TED(21)10:TP(TE2)(D(2)2)(1D(2))D(10):T1((TP)):T2T11:"@6V34H@"TP:16368,0:TP$C$}*HG$:43000:VT12:TE$" terminating "D$:43000:VT15:TE$" repeating "D$:43000:600(B"@18V2H@Then it is @I@IRRATIONAL@I@.")C340:PP1:"@I21H7V@ @IF@"7)"@2UL24H@"7)"@R@":HT2:VT9:BL36:LN10:350:P(LM1)4412:0:157,37157,67:158,37158,67((T$,14,1)"7"),46'>328:KK128:K82K734414:"@I7V21H@"(K)I$:TP$C$:I((T$,4,1)):II(K73)I(K82):ITP$"@I@WRONG@I@":NCNC(M4)'?T3$(T$,4):T$"@2H@Is it a":T1$", so it is @I@RATIONAL@I@."U(@"@7V23H@"TP$:I4419:VT9:TE$"n "00&:T$(91):H19:H2270:V136:V268:500:V2156:500:"@I5V2H@Press "T$"R] if the @D2H@"N$" is "R$"; @D2H@"T$"I] if "IR$": @I@":157,37157,67:158,37158,67:100'<16368,0:400:T$T$(TP(P)):"@6V24H@"(T$,(T$)4):(T$,3,1)"1"ē237,462317.":152,125174,125:4330%"@15V6H@"T1"@G@2@R@ = "T1T1" (too small)@D6H@"T2"@G@2@R@ = "T2T2" (too large)@2D6H@Therefore @UL@"(16)"@R18V@"TP" is @I@IRRATIONAL@I@.":124,141146,141%340:PP1:P94320:1100&0P91100:PM9:LM4:4410:410:120$"@9V2H@<3> However, if you find consecutive@D6H@"G$"s too small and too large@D6H@to be "S$"s, the actual@D6H@"S$" is not an "G$"@D6H@and is @I@IRRATIONAL@I@."%(TP(TETE))ĺ"@16V6H@"TE"@G@2@R@ = "TP", so @UL@"(16)"@R16V@"TP" is @I@RATIONAL@I@If the result is@D6H@too small, try a larger "G$".@D6H@If the result is too large, try@D6H@a smaller "G$"."$"@15V2H@<2> If you find an "G$" which can@D6H@be squared to obtain "TP", then@D6H@the "S$" of "TP" (that@D6H@"N$") is @I@RATIONAL@I@.":340:35D(2)2)(1D(2))D(10):HT2:T1((TP)):T2T11:VT5:LN2:BL36"350:VT8:LN11:350:"@5V2H@Determine whether this integer has a@D2H@"R$" square root: "TPk#"@8V2H@<1> Square an "G$" which you@D6H@guess will result in a "N$"@D6H@close to "TP". T3$(T3$,4):VT9:TE$"n "G$:43000:VT12:TE$" terminating "D$:43000:VT15:TE$" repeating "D$:43000:600:"@18V2H@Then it is @I@IRRATIONAL@I@."!340:PP1:P54312!H19:H2270:V136:V2156:500:V260:500B"400:TED(21)10:TP(TE2)(@I@RATIONAL@I@.":T2$"@2H@Then it is @I@IRRATIONAL@I@.": **PP1g LM4:100:I14:TP$(I)T$(TP(I)): 400:"@R6V22H@"13)"@U5B@ ":HT2:VT9:BL29:LN10:350:T3$TP$(P):"@6V22H@"(T3$,(T3$)4):((T3$,3,1))ē223,462177((T3$,4)"7..."),46!$" is also"]v"@2H@an "G$". Otherwise, the square@D2H@root is always an "IR$" "N$".":310uP81200:P44318? H19:H2270:V136:V2156:500:V260:500:"@5V2H@Determine whether this real@D2H@"N$" is "R$": ":T$"@2H@Is it a":T1$"@9H@, so it is as either a terminating or@D2H@a repeating (periodic) "D$".@D2HI@Ir"R$I$" "N$"s cannot be fully" t"@2H@expressed as "D$"s because they@D2H@neither terminate nor repeat.":600:V1108:V2148:500:"@14V2H@An "G$" has a "R$" square@D2H@root only if the "S@D6H@whether the "S$" is@D6H@"R$" is whether the root's@D6H@numerator is "R$". If it@D6H@is, then the "S$" is both@D6HI@"R$I$" and an @I@"G$I$"."!310hP4210,1200`rH19:H2270:V144:V2100:500:"@6V2H@Every "I$R$I$" "N$" can be@D2H@expressed "@D6H@and therefore is "R$".":340:350:"@9V2H@<3> We know that every "G$" can@D6H@be expressed as a "F$" with@D6H@a denominator of 1. This means@D6H@that its "S$" also can be" "@6H@expressed with a denominator@D6H@of 1. Therefore the test forn@D6H@factors).":340:LN10:VT10:BL36:HT2:350"@10V2H@<2> For the square to be an "G$"@D6H@the term B@G@2@R@ must be equal to 1,@D6H@therefore B must equal 1. This@D6H@means that the "S$" must@2D12H@A@D6H@equal "T$", which is an "G$" (A)@D12H@1"D2H@"G$". Follow this reasoning:@3D2H@<1> Assume that the "S$" is@D6H@expressed in lowest terms as:"1T$(10):"@13V6H@A@32H@A@G@2@RD6H@"T$", which makes the square "T$" ,@D6H@B@32H@B@G@2@R2D6H@also in lowest terms (A and B@D6H@are "G$"s without commo2HI@"IR$I$" "N$"s.":600:"@2H@"T1$"@2D3H@.";I19:D(9);::"...(etc. - no repeats.)@2D3H@.";:I19:D(9);::"...(etc. - no repeats.)":600:310eH19:H2270:V136:V268:500:"@5V2H@For the "S$" of an "G$" to@D2H@be "R$", it must also be an@"@16V35H@1@D3B@= "T$"@DB@6":600:340:0:500:H153:H254:500:154,102160,102:11,116268,116:3:HT2:BL36:LN6:VT5:350:VT13:350-"@6V2H@Decimals which @I@do not terminate or@D2H@repeat@I@ cannot be expressed as@D2H@"F$"s. Therefore, they are@D4,102160,102:600:H153:H254:V1116:V2155:500:"@15V3H@10X@D2H@- X@D3H@"T$T$T$T$"@D4H@9X":600H1149:H2150:500:"@15V9H@1.6666(etc.)@D8H@- .1666(etc.)@2D9H@1.50 = 9X@U9H@";:I112:T$;::600:"@17V25H@X = "T$T$T$"@U29H@1.5@2D30H@9":600lso show that @I@repeating@D2H@"D$"s@I@ (which have an infinite@D2H@"N$" of digits, but with a@D2H@repeating pattern) can also be@D2H@expressed as "F$"s, so are also@DI2H@"R$"@I@ "N$"s.":600c10,116270,116:"@2H@"T1$" Let X be .16(the 6 repeats)":15"@U3B@237@2D4B@1000":600:H2178:500:"@16H@"T1$"@2D17H@.5 = "T$T$"@UB@5@2D2B@10":600H2270:500:"@26H@"T1$"@D@39@D9B@.39 = "T$T$T$"@D3B@100":600:340:0:V1101:V2155:H1106:H2178:500:3:VT5:LN5:BL36:HT2:350:VT13:350"@5V2H@We can aWe can see that @I@terminating "D$"s@ID2H@("D$"s which have a finite "N$"@D2H@of digits) can be expressed as@D2H@"F$"s, and are therefore@D2HI@"R$"@I@ "N$"s.":600WT1$"@13V@EXAMPLE:":V1100:V2156:H2107:500:T$(10):"@2H@"T1$"@2D3H@.237 = "T$T$T$T$<0> RETURN TO CONTENTS"h"@I10H22V@WHICH (0-4) ?? @I@":MN0:MX4:300:Kİ412:26:(4)"RUNAM5.1":MK:P1:36320M,(36320M)1:400:4103:C,,4000M4100,4200,4300,4400P4110,4120,1200H19:H2270:V136:V292:500:"@5V2H@"@D2H@NO YES":D$"decimal":F$"fraction":S$"square root":G$"integer"rTP(36251):TPC(TP1)4:410:5000LC3:410M0:P0:400:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@11V10H@<3> EXAMPLE@10H13V@<4> SAMPLE PROBLEM@10H16V@):"@R@":D(X)((1)X)1:35339:I$"@I@":RP$"represented":W$I$"WRONG"I$", try again":C$I$"CORRECT"I$:R$"rational":IR$"ir"R$:N$"number"T$(15):I115:T$(I):I6T$(I)T$(I)"..."H:TE$(0)"@I2H@NO@I@":TE$(1)"@I6H@YES@I@":T4$Kp628:(PSMX)(K141)(K136)624:PSMXĺ"@"HTPSL"H@"((512PS));wqK141630:K136PSĹ511PS,32:PSPS1rKK128:K47K58K45ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX630t622 vIN$"":I0PS:IN$IN$((512I))::IN(IN$4:V2156:5008XI11500::K(16384):MK155321:[bTPD(4)5:TPTP(TP10)118:g2(TP10)(TP15)3(TP138)4(TP133):l35400,L(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32::16368,0nPSMXĺ"@"HTPSL"HI@"((512PS))I$;41430976,"@21V1HLI@"19)"@RI@":lH11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2: "@16V2H@Use "(91)(2)","(1)"] keys to light the correct@D2H@choice, and the "(91)"RETURN] key to@D2H@select your choice.":H19:H2270:V112Use the "N$" keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Use "(91)(2)"] to backspace and@D2H@"(91)"RETURN] when you finish each part." oH19:H2270:V1116:V2156:500 "@2H1V@"C"@5H@"P(P10)"@2HD@"M:3:31051:I$38)I$:I:2:38)::2 c***TOGGLE INVERSE***K eBL1:VV1:V2V1LN1 g"@"V"V"HT"HI@"T$(BL)I$:328:KYK:16368,0:"@"V"V"HT"H@"T$(BL):KY141ı iKY149VV1:BLBL1:VV2357 kKY136VV1:BLBL1:VV1VV2:BLLN l359 m"@15V2H@31 L63900. MK136412:K149410:1300n T"@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINUE@I@":16368,0 Y320:K160345:"@22V1HI@"36)"@I@": ^"@R0K"VT"V@";:I1LN:"@"HT"H@"BL)"":: `VT15:HT1:BL38:LN5:350 bI2051:I:2:(K136K149)(M4))305:K155ıL C"@40X40YN@";:328:K155K205ı` D16368,0:326k E:325w F63900 G412:K1551200:26:(4)"RUN ALGEBRA 5": H16368,0 IK(16384):K128329: JPP(K149)(P0)(K136):332 K:3d 6"@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE@I@"m 7320 8KK128:K27(36251)0323:K21K8311:K21PP1:410 >K8PP1:412:P1200 ?400:1300 @ZH(1):328:(36251)ı$ A(P%24577:3000063900%/ 1000C EWS3 ROUTINESV0:1002:::odTP(1)D(15):I2LMfTP(I)D(15):OK0:J1I1:OKOK(TP(I)TP(J))::(OK0)102::,320:KK176:KMNKMX300:13072:306363900 4312               IN AM5.1":S:O"@11V25HI@USE POSITIVE@D25H@VALUES ONLY.@I@":IN0:340:"@11V25HL@"6)"@R@"::PÃRATIONAL AND IRRATIONAL,ROOTS OF NUMBERS,PROPERTIES,IRRATIONAL NUMBERS TEST:(222)253Ħ ;:26::"++ ERROR ++";::" "(222):"AT LINE #"(218)(219)256:"020412:"@I11V25H@WRONG@I@,@D25H@try again.":340:"@L11V25H@"5)"@R@"9OT3$C$:620:(INTP)ĴIN020412:T3$"@I@WRONG@I@"9OT1$(Z):I2X:T1$T1$"*"(Z)::"@10V25H@"T3$",@D25H@"Y"="T1$",@D25H@so the @D25H@answer is "Z"."::O20420:20407N0:340:"@12V2H@"36):@8LO***RANDOM NUMBERS FOR C2M3 AND M4o8QOI02:J03:A(J,I)0:J,I:I145(M4)8VOX(I)D(3)1:Z(I)D(4)1:A(X(I)2,Z(I)2)20310:A(X(I)2,Z(I)2)1::8O ***WRONG ANSWER ROUTINE FOR C2M4G9OT3$C$:INTP20410:INONG@I@, see correct answer above.":600:"@12V2H@"32):20115o7N"@12V2H@"C$"."24):600:"@12V2H@"8):201157N20120:201057N35400,2(L2):"@"HT"H"VT"V@"MX)"@"H2"H"15123(L2)"C@"TP"@R15C@":8N"@12V2H@"5)I$"USE POSITIVE VALUES ONLY.@I@":IN$" line but cannot be expressed as@2D1H@an integer ratio. Therefore, it is:":TP2:v6 N***ROUTINES FOR CONCEPT 26N ***WRONG ANSWER ROUTINE FOR C2M1P2=7NINTPĴIN020120:"@12V2H@"W$:340:"@12V2H@"32):620:(INTP)ĴIN020112:"@I12V2H@WR9):TPX(X1):X(X1)X(X2):X(X2)TP::56XX(P):Y((X)):ZY1:TP1:X1X4X9T1$"precisely "(Y)", which can be written@2D1H@as an integer ratio. Therefore, we@2D1H@know that it is:":W67T1$"somewhere between "(Y)" and "(Z)" on the@2D1H@"MX3:620:TPZ:NCNC(C4)(INZ):20400H4Z 340:PPMPP1:400:3412N4[ 436251,1:P1:PM4:NC4:2410:36252,NC:P1:400:NC0:3410:36253,NC:26:(4)"RUNAM5.1-2":4'4'***ROUTINES FOR CONCEPT 1$56I19:X(I)I::I110:X1D(9):X2D(76:500:V2116:500:3653T XX(P):ZZ(P):Y(ZX):HT2:VT5:BL36:LN4:I$:350:VT10:I$:350:"@I5V2H@Find the value of:@D23HL@"(15)Y"@R23H@"X,I$:0:174,4717414((Y)),47(4V "@3H11VL@"(15)Y" =@R3H@"X:3:34,873414((Y)),87:HT17:VT11:L2:$"The "TX$(X)T3$" root of "(Y)" is "(Z)"."2 "@15CIR18V2H@"36)"@"20((T2$)2)"H@"T2$I$:340:P4PP1:HT2:VT5:BL36:LN4:350:VT11:LN8:350:400:33122 410:12002H P91100:PM9:3410:410:12003R 20300:H19:H2270:V136:V2a multiplication@D2H@factor "X" times, results in "Y".":600:T1$"*"(Z):TP$(Z):I2X:TP$TP$T1$::TP$TP$" = "(Y)1 T3$"nd":TX$"principal ":X2T3$"rd":TX$"":X3T3$"th":TX$"principal "62 600:610:"@"TP"CL15V"20(TP$)"H@"TP$:600:T241200:P13310:3312"0 203000 XX(P):ZZ(P):Y(ZX):H19:H2270:V136:V2156:500:V276:500:"@5V2H@Find the value of:@L6V23H@"(15)Y"@R23H@"X:174,4717414((Y)),47:600z1 "@11V2H@We want to find the positive "N$"@D2H@which, when used as quare@2D2H@roots@I@ ("N$"s which when raised to@2D2H@the second power result in the first"/ "@D2H@"N$"). Both have the same@2D2H@absolute value.":600:"@16V2H@The @I@principal root@I@ is a positive@2D2H@"N$". The other is negative.":600:3100 P X is written as:@2D11H@index >n@D4H@radical sign > X< radicand@UEL19H@"(16)"@ER@":114,131130,131:114,139130,139:156,139173,139:146,134154,134:310Z/ H19:H2270:V136:V2116:500:V1124:V2156:500:"@5V2H@Every positive "N$" has two "I$"s196,126203,126:310*- P3210,3220,1200- H19:H2270:V152:V292:500:V1108:V2148:500:"@7V2H@The "I$"root"I$" of a "N$" is a second@D2H@"N$" which, when raised to a@D2H@specified power, results in the@D2H@first "N$".". "@14V2H@The nth root of6H@root) of a negative "N$".@2D2H@<4> The square root of a "N$" can",B "@6H@be written as:@DE6HG@"(2)"@BL@"(16)"@R16V@X (with index) or @L15V@"(16)"@R16V@X@D6H@(without). All other roots must@D6H@have the index specified.@E@"-C 3:56,12663,126:ame absolute@D6H@value, but is negative.@D6H@This is true for all roots with@D6H@even integer indices.":340:"@L8V@";:I05D,A "@1H@"19)""::"@R8V2H@<3> There is no solution in the real@D6H@"N$" system for an even@D6H@integer root (such as a square@DN$" is a CUBE ROOT)."*= 340:"@L8V@";:I05:"@1H@"19)""::"@R8V2H@<2> Every positive "N$" has@D6H@exactly two square roots (not@D6H@necessarily integers or@D6H@"R$"), one of which is"}+> "@6H@POSITIVE (the PRINCIPAL ROOT).@2D6H@The other has the s> Numbers with an exponent or@D6H@index of 2 or 3 have special@D6H@names:"*< "@D6H@A "N$" to the second power is@D6H@SQUARED (and the second root of@D6H@a "N$" is a SQUARE ROOT).@2D6H@A "N$" to the third power is@D6H@CUBED (and the third root of a@D6H@":L1:620(8 TP4:H29:20100:HT11:MX3:L2:620:TP81:H211:"@12V2H@"32):20100:0:106,71119,71:3:"@L9V17H@=@R@":HT21:MX2(9 620:TP3:H221:"@12V2H@"32):20100:310J): H19:H2151:V136:V252:500:5:500:"@5V2H@Some special notes:@3D2H@<1als:@IL9V18H@=@R@":HT22:L2:MX4:620:TP81:H222:"@12V2H@"32):20100 (6 340:HT2:VT5:BL36:LN8:350:"@5V2H@We can write 3 to the 4th power as@D2H@3@G@4@R@. How would you write the 4th@D2H@root of 81?@L9V9H@"(15)"@R@":75,71119,71:HT8:VT9:MX2:HT1:BL38:LN16:350&4 365:V136:V2108:500:"@5V2H@To write 3 to the 4th power, we can@D2H@write 3*3*3*3, or our new notation:":HT9:VT9:MX3:L2:620:TP3:H213:20100:HT15:L1H'5 620:TP4::H215:"@12V2H@"32):20100:"@I7V22H@which @D22H@equot of a "N$" Y@D2H@(like the 4th root of 81) we write:@2D2H@'index' (n)@D2H@radical sign@25H@'radicand'(Y)"&3 610:615:"@16V16H@>@DB@>@5F@<@L16V17H"TP"C@"(15)"Y@R15C17H@n":132,127152,127:3:94,131117,131:101,139117,139:156,139174,139:340:VT4:500:"@5V2H@This is the notation we use to show@D2H@a "N$" X raised to the power n:@2D3H@We call X the@D3H@'base'...@5F@>":86,75124,75k%2 610:"@L8V19H"TP"C@X@R15C@n <":163,67187,67:"@8V27H@...and n is@D23H@the 'exponent.'@2H13V@To show the nth roH@resulting in the value 81.@2D1H@On the other hand, the "N$" 81 can"#, "@15V1H@be said to have 3 as its 4th root,@D1H@that is, the "N$" which can be used@D1H@as a factor 4 times to result in 81.":600:310$0 H19:H2270:V136:V292:500:V1100:V2156f@D1H@fact, we can also write 81 as@L14V6H"TP"C@3 * 3 * 3 * 3@R17V1H15C@because when we use 3 as a factor 4@D1H@times, the result is 81."E#* 340:VT10:HT1:BL38:LN9:350:"@10V1H@In this example, the "N$" 3 is said@D1H@to be raised to the 4th power,@D120,3130,1200!& H19:H2270:V136:V268:500:"@5V2H@Sometimes we describe one "N$" by@D2H@expressing it as some other "N$"@D2H@multiplied by itself.":600:"@10V1H@For example, to describe the "N$" 81""( 610:"@11V1H@we can write 9 * 9. As a matter o)::"@17V2H@"W$:NCNC(C4):600:510:HT1:V112:LN3:355:BLTPT1$I$"WRONG"I$ r I02:"@"16I"V2H@"35)::"@17V2H@"T1$", the square root of "X" is@D2H@"T2$"." s PPMİ340:PP1:2412 t 340:410: M3100,3200,3300,3400 ! P3110,31l 400:"@4V15K@";:I010:"@"1515(I6)"K1H@"38)""::14050:"@I4V1H@The "N$" which is the square root of@2D1H@"X" is "T1$I$n "@12V@";:I13:"@1H@"T$(I)::510:HT1:V112:LN3:355:T2$R$:TP2T2$IR$U p T1$C$:BLTPāI02:"@"16I"V2H@"35@2D21H@as the ratio of@2D21H@two integers, so@2D21H@it is "IR$"." 340:PP1:P5VT7:BL17:LN11:HT2:350:HT21:350:400:2312 410:1200` P91100:PM9:2410:1200j T$(1)R$".":T$(2)IR$".":T$(3)"possibly "R$" or "IR$".":14000V AN2312:Y(P,1)Z:Y(P,2)X:"@7V12H@"X0)X"@D2H@The "N$" "(10)" is@D12H@"Y0)YA "@11V2H@a ratio of two@2D2H@integers, "X" and @2D2H@"Y", so it is said@2D2H@to be "R$".":600:"@7V21H@There is no way@2D21H@to express the@2D21H@square root of "Z" P41200~ V136:V2156:H19:H2137:500:H1143:H2270:500:I$"@5V2H@"17)"@2F@"17)"@5V6H@RATIONAL@24H@IRRATIONAL@I@" XD(18):YD(18):XX9(X10):YY9(Y10):XY(XY)2312:ZD(6):ZZ1(Z2):AN0:P1āI1P1:ANY(I,1)ZY(I,2)XAN:@The set of "I$R$I$" "N$"s@D2H@includes all real "N$"s which can@D2H@be "RP$" as the ratio of two@D2H@integers (that is, as fractions).":600"@13V2H@The set of "I$IR$I$" "N$"s@D2H@includes all real "N$"s which@D2H@cannot be "RP$" as fractions.":310P"H@"T$(I)::"@6V"TP10"H@"T2$"@3D2H@"T3$RP$" as an integer ratio.":510:HTTP:V15:LN3:355Q"@12V2H@"T2$", a"T1$" "N$" "T$(1(TP20))"@D2H@can be "RP$" as a ratio of two@D2H@integers.":352P2210,1200H19:H2270:V160:V2132:500:"@8V2H"@12V2H@"17):355:BL2T2$I$"WRONG"I$L2129:340:HT2:BL36:LN6:VT5:350:VT12:350:T1$"n "IR$:T2$"can":T3$"be ":TP23:2128:T2$C$:BL1ĺ"@12V2H@"W$:355:BL1T2$I$"WRONG"I$N2129:310^P"@6V2H@A"T1$" "N$" @5V@";:I13:"@"Tquare root of 2 an@D2H@"I$IR$I$" "N$" because it cannot@D2H@be "RP$" fully in ratio form.":310HT$(1)"never":T$(2)"always":T$(3)"sometimes":H19:H2270:V136:V292:500:T1$" "R$:T2$"can be":T3$"":TP20:2128:T2$C$+JBL2ĺ"@12V2H@"W$:340:D160,41157,44160,47:174,41177,44174,47:"@10V2H@Somewhere in the range shown above@D2H@is a "N$" (the square root of 2,@D2H@which we'll describe later) which@D2HI@cannot@I@ be "RP$" as a ratio of@D2H@two integers."_F600:"@16V2H@We call the s "N$"@D2H@because it can be "RP$" as the@D2HI@ratio@I@ of two integers."B"@D2H@For example: 2 4 6@D15H@2 = "(10)" = "(10)" = "(10)" etc.@D19H@1 2 3":340:350:"@5V22H138C@"(9)"@2F@"(9)"@ED22H@"(9)"@2F@"(9)"@15CE@":7:157,44177,4410:J43I:"@"J"H6V@"(9)"@"J(I5)"H7V@"I5::"@6V2H@<@37H@>":6:15,51262,51:3:H19:V176:H2270:V2156:500:HT2:VT10:LN9:BL36H@"@5V25H@X@10V2H@The X marks the representation of@D2H@the "I$R$I$" "N$" 2 on the "N$"@D2H@line. We call 2 a "R$"XAMPLE@10H13V@<4> SAMPLE PROBLEM@10H16V@<0> RETURN TO CONTENTS""@I10H22V@WHICH (0-4) ?? @I@":MN0:MX4:300:Kİ412:1100MK:P1:36320M,(36320M)1:400:4103:C2000,3000,,5000M2100,2200,2300,24004P2110,2120,1200>I00:MX4:300:CK:400:T$C$(K):C1T$T$" NUMBERS"ue:24(T$)2:"@2VI@"T$I$:Cİ412:26:(4)"RUNALGEBRA 5":gC3İ26:(4)"RUNAM5.1-2"j410:C41300?M0:P0:400:"@14H5V@LEARNING MODE@10H7V@<1> DISCUSSION@9V10H@<2> RULE@11V10H@<3> E,138:"@16V3H@<0>"[59,3259,159:60,3260,159:"@20H5V@CONTENTS@6V@":I14:"@9H@<"I"> "C$(I):I1I3ĺ"@13H@NUMBERS":I3ĺ"@2U24H@OF IRRATIONAL@D@"`:I:"@9H18V@<0> RETURN TO ALGEBRA MENU":"@2V7HI@"31)"@11H21V@WHICH ONE (0-4) ?? @I@"4cMN"CORRECT"I$:R$"rational":IR$"ir"R$:N$"number"LP0:M0:C0:400:H117:H245:I03:V14516I:V2V112:500:"@3H"62I"V@<"I1">":V14516(I1)16(I3)V32,V232,V136,V13:31,V231,V127,V13::20,13813,13220,12642,12649,13242,138208K45ĺ"@"HTPSL"H@"(K);:512PS,K:PSPS1:PSMX630At622v(PS0)620:IN$"":I0PS:IN$IN$((512I))::IN(IN$):"@R@":2D(X)((1)X)1:35339:C$(4):I14:C$(I)::I$"@I@":RP$"represented":W$I$"WRONG"I$", try again.":C$I$l35400,L(L1):"@"VT"V"HT"H@"MX):PS0:I0MX:512I,32::16368,0rnPSMXĺ"@"HTPSL"HI@"((512PS))I$;p328:(PSMX)(K141)(K136)624:PSMXĺ"@"HTPSL"H@"((512PS));qK141630:K136PSĹ511PS,32:PSPS18rKK128:K47K5)"] keys to light the correct@D2H@choice, and the "(91)"RETURN] key to @D2H@select your choice.":H19:H2270:V1124:V2156:500XI11500::K(16384):MK155321:bTPD(4)5:TPTP(TP10)118:g2(TP10)(TP15)3(TP138)4(TP133):Gen you finish each part."< oH19:H2270:V1116:V2156:500f "@2H1V@"C"@5H@"P(P10)"@2HD@"M:3:v 31051:414 30976 "@21V1HLI@"19)"@RI@": H11,V2H11,V1H2,V1H2,V2H1,V2H1,V1:H21,V1H21,V2:"@16V2H@Use "(91)(2)","(1L):KY141ı0 iKY149VV1:BLBL1:VV2357[ kKY136VV1:BLBL1:VV1VV2:BLLNd l359 m"@15V2H@Use the "N$" keys to fill in your@D2H@answer where the lighted cursor is@D2H@showing. Use "(91)(2)"] to backspace and@D2H@"(91)"RETURN] whE@I@":16368,0; Y320:K160345:"@22V1HI@"36)"@I@":k ^"@R0K"VT"V@";:I1LN:"@"HT"H@"BL)"":: `VT15:HT1:BL38:LN5:350 c***TOGGLE INVERSE*** eBL1:VV1:V2V1LN1 g"@"V"V"HT"HI@"T$(BL)I$:328:KYK:16368,0:"@"V"V"HT"H@"T$(B6 E:325 F63900I G412:K1551200:26:(4)"RUN ALGEBRA 5":W H16368,0t IK(16384):K128329: JPP(K149)(P0)(K136):332 K:331 L63900 MK136412:K149410:1300 T"@22V6HI@PRESS "(91)"SPACE BAR] TO CONTINU 7320K 8KK128:K27(36251)0323:K21K8311:K21PP1:410h >K8PP1:412:P1200w ?400:1300 @ZH(1):328:(36251)ı A(P(K136K149)(M4))305:K155ı C"@40X40YN@";:328:K155(K205C)ı D16368,0:32 324577:3000063900) 1000= EWS3 ROUTINESP0:1002:::r,320:KK176:KMNKMX300:|13072:3063639004312 6"@3H21VI@PRESS "(1)" KEY TO VIEW THE NEXT PAGE@D3H@PRESS "(2)" KEY TO VIEW THE LAST PAGE@I@"               "P$:35339:VPÃIRRATIONAL NUMBERS,RADICAL EXPRESSIONS,FINDING SQUARE ROOTS,POSTTESTh(222)253Ħ:26::"++ ERROR "(222)"++"::"AT LINE #"(218)(219)256:"IN ALGEBRA 5": %EXAMPLE":A$(4)"SAMPLE"[TFI14:9:N:A$(I);:28:(L(I)100M);:NN2:31:"%"::24: N26:(4)"OPEN"P$:(4)"READ"P$:I14:C(I)::(4)"CLOSE":35339: R26:(4)"UNLOCK"P$:(4)"OPEN"P$:(4)"WRITE"P$:I14:C(I)::(4)"CLOSE":(4)"LOCK9H3V@"(23)(24)"@10H9V@"(22):I14:"@"63I"V8H@<"I">";:I1ĺ"@U2B@"(22)V.:PF::26:M0:I14:L(I)(36320I):MML(I)::MĿRF:I124:I:40):::I521:I:5:32)::16:6:"MODE USAGE":N9:A$(1)"DISCUSSION":A$(2)"RULE":A$(3)",54,185:275,5275,185:3,25276,25:3,165276,165:45,545,25:46,546,25.7,167272,167:"@I@":I12:I1:"@7H@"32):I21:"@1H@"38):.24(U$(K))2:"@1V@"U$(K)"@I1V1H@C0 P0@D1H@M0".49,7272,7:N."@5V3H@<9>< MENU > <0>@7H2V@START@2742,1949,11:49,4370,3291,4370,5649,43:16,4916,3742,3742,4917,4917,37:43,3743,49+69,3269,28:45,4349,43:91,4393,43:104,37124,37133,43124,49104,4996,43104,37:66,6568,6872,65:K.:35339:3:3,5276,5276,1853,1853,5:43> FINDING SQUARE@D24H@ROOTS"n'140,7272,7:"@16V20H@<4> POSTTEST@17V20H@<9> RESET MENU@18V20H@<0> STOP":*YY24:3:66(I1)3,Y166(I1)3,Y11:C:X,Y12X,YX25,YX25,Y12X1,Y12X1,Y:X24,YX24,Y12:_+49,1184,1191,1984,2749,137,5137,186G'"@I@":I13:I1:"@20H@"19):I20:"@20H@"19):'"@26H1V@ALGEBRA@25H2V@VOLUME #5@20H3V@VER "VR$" "DA$"@21H21V@WHICH (0-9) ??@I25H 5V@CONTENTS@20H7V@<1> IRRATIONAL @24H8V@NUMBERS@10V20H@<2> RADICAL@D24H@EXPRESSIONS@20H13V@<1C(Q)5!W0C(Q)1:C(Q1)6K21000:400:36251,0:W0QQ1:2060d I112:36250I,0: Q31010:KQ:12000:26:(4)"RUN AM5."Q'3:3,5276,5276,1863,1863,5:4,54,186:275,5275,186:136,5136,186:136,34276,34:136,157276,157:TIONS. YOU HAVE PASSED@D2H@THE POST TEST AND MAY NOW GO ON TO@D2H@ALGEBRA 6.@I@":2050TP$(0)"UNIT "(Q1):TP$(1)"THE POST TEST":"@I15V2H@CONGRATULATIONS. YOU HAVE PASSED@D2H@UNIT "Q" AND MAY NOW GO ON TO@D2H@"TP$(Q3)".@I@"W1C(Q)6W13(XX2)(XX3):R(I)"@6F@"4R(I)"@D23H@";::"@I138K8V34H@";:I13(XX2)(XX3):R(I)3ĺ"@G@"(124)(125)(125)"@R@";:WW1Q(36251):"@D34H@";:I:"@128KI@":W0ĺ"@I14V2H@ARROWS SHOW AREAS OF WEAKNESS.@I@":2050VQ4ĺ"@I15V2H@CONGRATULA40x "@8V2H@APPROXIMATES@D2H@COMPUTE ROOTS@D2H@USING TABLES@D2H@PYTHAG. THEOREM@D2H@";:I14:R(I)(36258I)::2040 "@8V2H@IRRATIONAL NUMBERS@D2H@RADICAL EXP.@D2H@SQUARE ROOTS@D2H@";:I02:R(I1)(362544I):W0:"@8V23H@";:XX(36251):I30,2032,2034,2036 "@7V2H@RATIONAL AND@D2H@IRRATIONAL@D2H@ROOTS OF NUMBERS@D2H@IRRATIONAL NUMBERS@7V23H@";:I13:R(I)(36251I)::2040 "@8V2H@SIMPLIFYING@D2H@ADD. AND SUB.@D2H@CONJUGATES@D2H@RADICAL EQUATIONS@D2H@";:I14:R(I)(36254I)::200,110228,110:144,49144,110:143,49143,110:186,49186,110O 13,39265,39 185,49185,110:228,49228,110:227,49227,110:"@5V2HI@"36):I14:3:14I:36):I:I18:34:6I:5): "@I5H1V@0@I5V8H@CONCEPT@21H@RIGHT WRONG@I@":(36251)20:K4Ĺ36251,4:24576,0:I115:36251I,0::(4)"RUNAM5.1-2"R (4)"RUNAM5."K  TEST REPORTING//PEEK(36251)=UNIT #.//? TEST RESULTS AND UPDATE MENU COLORS 3= 10,37268,37268,14510,14510,37:11,3711,145:269,37269,145:10,49228,49:1160400:"@I22V1H@"36)"@I@":D 20000:35339:((36251)0)2000q :10000:12010:300:3:11010:16368,0 K(16384):K1281020 16368,0:KK176:K18000 K9C(1)6:I24:C(I)3::300:21000:1020 K1K41020> 12000:2624577:30000:63900[I14:U$(I)::VR$"1.2":DA$"01 MAY 84":P$"AM5.PROGRESS"e 1000x0:1002:::,X53:Y45:I14:CC(I):11000::"@22V6HI@PRESS SPACE BAR TO CONTINUE@I@":16368,0 K(16384):K128401:16368,0:K              " x@"  c <`@x   6 p"p>>"><"">"","">> >"<,<"*" ">@pp@pp@p@99>1,>>< ?~>``~x> ><,<"*" ">~ c>> "8x2 "" "" >""""""2" "-"0""&2" *"", "*"@@@@P@P@)!> 2*A~& ~>`~0|xw>0"&2" *"", "*0{"> "<x`>, & > "" >>""2" "2"* ""-0"<&2>"*""&2""*A@ppppp@pp)9(,]~*@~0>>x`|xxx0<&2>"*""&2""*~?o"> "8>x*`9R >* <>"">**""""!0Ar &2"& *&"&2&""*@@@P@PP0) ?" >E<>2@< >x`x p xx0 &2"& *&"&2&""*0>c"> p"pxI0>R 2 "" >"""" *&""""""!"0"",,<>"""">@ppPpppp99"  ]<@>`>`p` x>0",,<>""""> 6>> " """6&""""""!"" 0  &<& *A@@ >``@ x0    "xAW6 >>8>>>><""""<>""!"">>> 8  6 >@0 >`@ x>> 8 D "LU :F`F`$L"%e%`$e($`%80%`$80$`'$L:{|0L_`F)׭F F)L i)`) qp`<) Ji L? 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((d d22(LI GIG`RH`GH`LH`E yIy`OF) ELSSzI zU`TR`P EI E`NLUX F{LY F|LTIF eiLȱ|ȱ{ U`FFB DLF .L`% ImE8Ie$e΅ϩeeυ GIy) e$) =}м`0:)F FmFeF`@D`C r)׍} ~  `K r)׍ `H F($LV F%@ JTL J UEPWRz6F GFy 7 ?` !!""## ( !(!"("#(# P !P!"P"#P# ((ˍ) J ?`DLk@D` LH5Ω " .Ωz % $L.jjΩ .@0~~@@|@>f@@@@ 0 0@pp`@@@@@@@@p@0`x@@@0@@@p&@|Lx0`@ @0| @0|| @L << @ 00 0~f>>fff@@@@@@ |@@@@@@@pp`p@0@ x@ pp|||p0pp@0@ |p|| p pp | p p |@p@ @|||@|xp@@@@@@p@ ||@@``@|@@@b@|L`0@ @0 0|| @L << @ 0 0|xff@@@@@@ |@@@`@xp@@@?p@x@0| ~@x@ |p||||0|p0@| |||| | p| | | | |@|@ @|||@~xx@@@@@@|@ |@@`@x@@@&@0L0@ @@ < @@ < @ < @ 0 0p|||||||@@@@@@@@@p@@~pp@X@?p@|0| p`@ <@ 0 @@@| @ @ 0< @ ```````@@@@@@ 0<00@@@@p@@?d@00|p`@ p @ 0 @@ 0 @ @ xxxxxxx@@@@@@@@@x`x@@f|p||@`~0@00@0<@ 0 @@@@| @ @ 0<0@  xxxxxxx@@@@@@ 0<00@@@@`@? @00px@x|p|||p~||0||p|||||| p |||||| || |0```````@@@@@@@0```|@`|||xp`p|@|@00@0l@<| 0 @0|@| @L LL@ L0@0 ~ |||||||@@@@@@ L@p@@@@@@8@00@`p@|||@~pp0p|@|p|||p p p|p|p| || |@@@@@@@@@0`x@@`~|x`p8 <`@|@00`@0l@0| 0 0|@|| @L L L@ L0@0 | f>@@@@@@ L@|@@@`@@@@@@@@~~@@|`~pppp @0p@`||@L@@@ |@pp@ ||| || |@< LL | |p@ @@@ @x ff>>fff@@@@ 0 0`?? 030>>0<0~131 p33>>?3?~ ?#33080<0  2  g~0 '~0 <?<3>8<<0x` < 77???630>>33?``y$%GΩϩ  %%%GΩϩ  %%$$'к`&$%GΩϩ  %%%GΩϩ  %%$$м`  p 4114440 400490 9009958 62 6226660125 125512211126 126612211124 1244122111256 667 """""""!!! ! ! ! ! !!!!!!!!!   LINE "(218)(219)256": EDU-WARE":::T;::" "(222):" AT LINE#"(219)256(218):%,B6%,E7%,B7%,E8%,B8%,E9%,B9%,E0%,B0%,F1%iÇC1%,F2%,C2%,F3%,C3%,F4%,C4%,F5%,C5%,F6%,C6%,F7%,C7%,F8%,C8%,F9%,C9%,F0%,C0%,G1%,D1%,G2%:(222)255Ħ'80::::"++ ERROR ++"::"ERROR "(222)" AT 126,112188,112:140,81160,81:126,111188,111o 3:167,83:173,87:179,82:185,84:175,91:183,90:185,97 "@R15C15H15V@2@20H@2@25H@2@L16H15V@=@21H@+@13H@"(97)"@18H@"(98)"@23H@"(99) x X(222):X255Ħ.:0:1002::::"++ERROR++" 171,53169,55167,56166,58165,60164,64164,66165,68166,71167,73169,75171,77 178,67188,56189,56179,67:"@R27H6V@"(6) "@IR15C@":VT1012:"@"VT"V23H@ "::"@I@" 6:A80:I161189.9:I,A189,A:AA1::30 140,80161,80:66,35:67,36214,36214,14867,14867,36a 1:I7791:I,48I42,96::I7791:I,96I42,48: 5:141,48196,72141,72141,48:142,49196,71142,71142,49 7:170,53168,55166,56165,58164,60163,64163,66164,68165,71166,73168,75170,77WI:2:38)::"@I@"- `"@LI1V12H@ALGEBRA 5"U j"@R3V1H@VERSION 1.2@30H@01 MAY 84" t"@19V9H@Copyright @G@,@R@ 1983,1984@D11H@Peachtree Software@D13H@An MSA Company@22V10H@All Rights Reserved@I@"  PRINT PICTURE( 66,35213,35213,14766,147 "@R@"9 &I112:36250I,0::I3629936351:I,0:Z X26:80:(4)"RUN ALGEBRA 5"p  PRINT BOARDERS 4,4277,4277,1884,1884,43,4276,4276,1883,1883,4 L PRINT HEADERS WITH A BACKGROUND V"@RI@":I24:I:2:38)::I2023:0+205,255::::255:ZZ(0)::6390071012,0h COPYRIGHT 1983 EDU-WARE SERVICES, INC.} 24576:27903:3 (4)"BLOADEWS3" 353395000:1002:P:1002:51,0::::35328:35397,32:230,64:1000:35339""" " " " " ""