,*APT] #H3 ,AN ,9TRODUC;N    #H4A
    TO ,NEME? ,CODE ,SYMBOLS ,US$ 9
      ,ALGEBRA #A1 ,ALGEBRA #B1 &
           ,ADV.ED ,MA!MATICS

    ,NEW ,SYMBOLS ,9TRODUC$
.=\ .<DOTS #A-#B-#E-#F.> ,V]TICAL B> US$
  ) S* T1 MAGNITUDE1 NORM1 & DET]M9ANTS
.=,= .<DOTS #A-#B-#C-#D-#E-#F.> ,9F9;Y
  SYMBOL
.=.* .<DOTS #D-#F1 DOTS #A-#F.> ,HOLL[
  DOT
.=^^ .<DOTS #D-#E1 DOTS #D-#E.>
  ,SUP]SCRIPT ) SUP]SCRIPT 9DICATOR
.=; .<DOTS #E-#F.> ,SUBSCRIPT 9DICATOR
.=< .<DOTS #A-#B-#F.> ,9DEX-(-RADICAL
  9DICATOR

      ,GR\P+ ,SYMBOLS
.=..( .<DOTS #D-#F1 DOTS #D-#F1 DOTS
  #A-#B-#C-#E-#F.> ,OP5+ "<LEFT"> ANGLE
  BRACKET
.=..) .<DOTS #D-#F1 DOTS #D-#F1 DOTS
  #B-#C-#D-#E-#F.> ,CLOS+ "<"R"> ANGLE
  BRACKET                             #A
               ,*APT] #H           A#H4A

      ,5L>G$ ,GR\P+ ,SYMBOLS
.=,( .<DOT #F1 DOTS #A-#B-#C-#E-#F.>
  ,5L>G$ OP5+ "<LEFT"> P>5!SIS
.=,) .<DOT #F1 DOTS #B-#C-#D-#E-#F.>
  ,5L>G$ CLOS+ "<"R"> P>5!SIS
.=@,( .<DOT #D1 DOT #F1 DOTS
  #A-#B-#C-#E-#F.> ,5L>G$ OP5+ "<LEFT">
  BRACKET
.=@,) .<DOT #D1 DOT #F1 DOTS
  #B-#C-#D-#E-#F.> ,5L>G$ CLOS+ "<"R">
  BRACKET
.=.,( .<DOTS #D-#F1 DOT #F1 DOTS
  #A-#B-#C-#E-#F.> ,5L>G$ OP5+ "<LEFT">
  BRACE
.=.,) .<DOTS #D-#F1 DOT #F1 DOTS
  #B-#C-#D-#E-#F.> ,5L>G$ CLOS+ "<"R">
  BRACE
.=,\ .<DOT #F1 DOTS #A-#B-#E-#F.> ,5L>G$
  V]TICAL B>



                                      #B
               ,*APT] #H            #H4B

      ,SET ,!ORY
.=_0 .<DOTS #D-#E-#F1 DOTS #C-#E-#F.>
  ,EMPTY SET "<NULL SET">
.=.+ .<DOTS #D-#F1 DOTS #C-#D-#F.>
  ,UNION
.=.% .<DOTS #D-#F1 DOTS #A-#D-#F.>
  ,9T]SEC;N
.=_"K: .<DOTS #D-#E-#F1 DOT #E1 DOTS
  #D-#F1 DOTS #A-#E-#F #A.> IS A SUBSET
  (
.=@E .<DOT #D1 DOTS #A-#E.> ,IS AN ELE;T
  (

      ,GREEK ,SYMBOLS
.=.A .<DOTS #D-#F1 DOT #A.> ,GREEK LR
  ALPHA "<L[]CASE">
.=.B .<DOTS #D-#F1 DOTS #A-#B.> ,GREEK
  LR 2TA "<L[]CASE">
.=.? .<DOTS #D-#F1 DOTS #A-#D-#E-#F.>
  ,GREEK LR !TA "<L[]CASE">
.=.,D .<DOTS #D-#F1 DOT #A1 DOTS
  #A-#D-#E.> ,GREEK LR DELTA
  "<UPP]CASE">                        #C
               ,*APT] #H           A#H4B
.=.,S .<DOTS #D-#F1 DOT #A1 DOTS
  #B-#C-#D.> ,GREEK LR SIGMA
  "<UPP]CASE">

    ,NEW ,,BANA ,T]MS
_4 ^1,HOLL[ ^1DOT3 ,! HOLL[ DOT HAS BE5
  US$ 2F )9 ! DEGREE SYMBOL4 ,9 ? *APT]1
  X IS US$ Z AN OP]A;N SIGN REPRES5T+ !
  COMPOSI;N ( TWO FUNC;NS4

    ,KEY ,PO9TS
_4 ,A NUMB] 5CLOS$ )9 V]TICAL B>S IS US$
    TO REPRES5T ^1ABSOLUTE ^1VALUE4 ,!
    NUM]IC 9DICATOR IS N US$ :5 PLAC+ A
    NUMB] 2T TWO V]TICAL B>S4
_4 ,! ^19F9;Y ^1SYMBOL IS US$ TO
    REPRES5T T A VALUE 3T9UES ="E 9 !
    POSITIVE DIREC;N4 ,= NEGATIVE 9F9;Y1
    ! VALUE 3T9UES ="E 9 ! NEGATIVE
    DIREC;N4
_4 ,:5 WRIT+ ^1FUNC;N ^1NOTA;N1 P>5!SES
    >E US$4
_4 ,:5 WRIT+ ^79T]VAL NOTA;N1 ^' P>5!SES
    &_/OR BRACKETS >E US$ ) A COMMA   #D
               ,*APT] #H           B#H4B
    2T ! 2G9N+ ( ! 9T]VAL & ! 5D ( !
    9T]VAL4 ,A P>5!SIS M1NS ! VALUE IS N
    9CLUD$ 9 ! 9T]VAL & A BRACKET M1NS !
    VALUE IS 9CLUD$ 9 ! 9T]VAL4
------------------------------------#H4C
_4 ,! ^1HOLL[ ^1DOT IS US$ = DEGREES &
    AL = COMPOSI;N ( FUNC;NS4
_4 ,9 ADDI;N TO ! GR\P+ SYMBOLS (
    P>5!SES & BRACKETS1 ^1ANGLE
    ^1BRACKETS >E A TYPE ( GR\P+ SYMBOL
    MO/ COMMONLY US$ 9 MA? RELAT$ TO
    VECTORS4
_4 ,! ^1V]TICAL ^1B> C 2 US$ 9 DI6]5T
    WAYS 9CLUD+ REPRES5T+ 8S* T0 9 SET
    NOTA;N1 ! 8NORM (0 9 VECTORS1 OR !
    8DET]M9ANT (0 9 MATRICES4
_4 ,GR\P+ SYMBOLS S* Z P>5!SES1
    BRACKETS1 BRACES1 & V]TICAL B>S C 2
    5L>G$ & >E COMMONLY US$ 9 MATRICES1
    DET]M9ANTS ( MATRICES1 & SY/EMS (
    EQUA;NS4
_4 ,PROBLEMS 9VOLV+ 5L>G$ GR\P+ SYMBOLS
  _4 ,>E BRLD Z SPATIAL >RANGE;TS1 S
    L1VE A BLANK L9E ABV & 2L !       #E
               ,*APT] #H           A#H4C
    >RANGE;T4
  _4 ,M/ H AT L1/ "O 5L>G$ GR\P+ SYMBOL
    ON EA* L9E ( ! >RANGE;T4
  _4 ,M/ H ! 5L>G$ GR\P+ SYMBOLS L9$ UP
    V]TICALLY4
  _4 ,"S"TS H PROBLEM ID5TIFI]S1 "PS (
    AN EXPRES.N1 PUNCTUA;N1 ETC4 3NECT$
    TO ! >RANGE;T4 ,^! M/ 2 PLAC$ ON !
    TOP L9E ( ! >RANGE;T 9 BRL REG>D.S (
    XS LOCA;N 9 PR9T4
_4 ,"! >E MULTIPLE SYMBOLS US$ 9 SET
    !ORY 9CLUD+ ! ^7EMPTY SET SYMBOL^'
    OR ^1NULL ^1SET1 ! ^1UNION SYMBOL1 !
    ^19T]SEC;N SYMBOL1 ! 8^7IS A SUBSET
    (0^' SYMBOL1 & ! 8^7IS AN ELE;T (0^'
    SYMBOL4
_4 ,! ^1UNION SYMBOL & ! ^19T]SEC;N
    SYMBOL >E SIGNS ( OP]A;N S "! IS NO
    SPACE ON EI SIDE ( ! SYMBOL4
_4 ,! 8^7IS A SUBSET (^'0 SYMBOL & !
    8^7IS AN ELE;T (^'0 SYMBOL >E SIGNS
    ( COMP>ISON S A SPACE M/ 2 LEFT ON
    EI SIDE ( ^! SYMBOLS4
_4 ,:5 AN EXPON5T "<^1SUP]SCRIPT">    #F
               ,*APT] #H           B#H4C
    IS N FOLL[$ BY A SPACE OR N AT ! 5D
    ( A L9E1 A ^1BASEL9E ^19DICTOR IS
    US$ TO RETURN TO BASEL9E AF WRIT+ !
    SUP]SCRIPT4
_4 ,:5 A ^1SUP]SCRIPT ^19DICATOR OR
    ^1SUBSCRIPT ^19DICATOR IS US$1 & !
    SUP]SCRIPT OR SUBSCRIPT IS IMMLY
    FOLL[$ BY A MA!MATICAL SYMBOL1 S* Z
    A SIGN ( OP]A;N1 GR\P+ SYMBOL1
    T]M9A;N 9DICATOR1 OR FRAC;N L9E1 !
    ^1BASEL9E ^19DICATOR IS US$ TO 5D !
    SUP]SCRIPT OR SUBSCRIPT & RETURN TO
    BASEL9E4
_4 ,NO BASEL9E 9DICATOR IS NE$$ IF A
    SPACE OR O!R LEVEL 9DICATOR FOLL[S A
    SUP]SCRIPT OR SUBSCRIPT4
_4 ,D N USE A ^1SUBSCRIPT ^19DICATOR OR
    A ^1BASEL9E ^19DICATOR :5 A V>IABLE
    HAS A NUM]IC SUBSCRIPT4
------------------------------------#H4D
_4 ,! ^79DEX ( A RADICAL 9DICATOR^' IS
    US$ TO %[ A RADICAL ) AN 9DEX S* Z A
    CUBE ROOT1 F\R? ROOT1 ETC4
_4 ,COMMON ^1FUNC;NS 9CLUDE S91       #G
               ,*APT] #H           A#H4D
    COS1 TAN1 LOG1 & LN4 ,REG>D.S ( :E!R
    "! IS A SPACE 9 PR9T 2T ! FUNC;N "N
    & :AT COMES AF X1 "! IS ALW A SPACE
    9 BRL 2T ! FUNC;N "N & :AT COMES AF
    X4 ,D N USE AN ,5GLI% LR 9DICATOR 9
    A FUNC;N4
_4 ,COMMON ^1,GREEK ^1LRS 9CLUDE ALPHA1
    2TA1 & !TA : >E L[]CASE & DELTA &
    SIGMA : >E UPP]CASE4 ,"! >E _M O!R
    ,GREEK SYMBOLS US$ 9 MA!MATICS4 ,A
    COMPLETE LI/ ( ,GREEK SYMBOLS C 2
    F.D 9 ! APP5DICES4

    ,9TRODUC;N
  ,9 ,ALGEBRA #A & ,ALGEBRA #B /UD5TS
LE>N AB EXPRES.NS1 EQUA;NS1 9EQUALITIES1
TABLES1 & GRAPHS4 ,9 ^! CLASSES1 /UD5TS
LE>N TO "W ) DI6]5T TYPES ( FUNC;NS S* Z
L9E>1 QUADRATIC1 POLYNOMIAL1 ABSOLUTE
VALUE1 PIECEWISE-DEF9$1 EXPON5TIAL1
LOG>I?MIC1 RA;NAL1 RADICAL1 &
TRIGONOMETRIC FUNC;NS4 ,^! FUNC;NS >E
US$ TO REPRES5T RELA;N%IPS 2T
QUANTITIES4                           #H
               ,*APT] #H           B#H4D

    ,ABSOLUTE ,VALUE
  ,Z EXPLA9$ 9 ,*APT] #F1 /UD5TS >E
9TRODUC$ TO ABSOLUTE VALUE 9 SIX? GRADE
"<,COMMON ,CORE ,/ATE ,/&>DS ,9ITIATIVE1
#BJAJ2 ,M>YL& ,/ATE ,DE"P;T ( ,$UCA;N1
#BJAE">4 ,ABSOLUTE VALUE IS REPRES5T$ BY
A V]TICAL B> ON EA* SIDE ( A NUMB] OR
EXPRES.N4 ,X TAKES "O CELL TO WRITE EA*
^1V]TICAL ^1B> 9 ,NEME? ,CODE & IS
WRITT5 ) DOTS #A-#B-#E-#F4
  .=\
  ,! V]TICAL B> = ABSOLUTE VALUE IS
3SID]$ A GR\P+ SYMBOL 9 ,NEME? ,CODE4 ,A
NUMB]1 :E!R POSITIVE OR NEGATIVE1 5CLOS$
)9 V]TICAL B>S T >E US$ = ABSOLUTE
VALUE1 DOES N NE$ A NUM]IC 9DICATOR
"<,RULE ,,II ^S#AA;E"<#H">">4

_% \5-8\ .K #3 _:
  ,R1D3 ,! ABSOLUTE VALUE ( FIVE M9US
    EI<T CLOSE ABSOLUTE VALUE EQUALS
    ?REE4
                                      #I
               ,*APT] #H            #H4E
  ,IF TWO SETS ( V]TICAL B>S >E PLAC$
NEXT TO EA* O!R1 A MULTIPURPOSE 9DICATOR
"<DOT #E"> M/ 2 PLAC$ 2T ! TWO SETS
"<,RULE ,,XXIII ^S#AGGVII">4
  _% \5\"\-3\ .K #15 _:

    ,ADDI;NAL ,USES ( ! ,V]TICAL ,B>
  ,"! >E O!R USES ( ! V]TICAL B>
9TRODUC$ 9 HI< S*OOL />T+ 9 ,ALGEBRA #A1
B ! V]TICAL B> IS ALW WRITT5 ) DOTS
#A-#B-#E-#F "<,COMMON ,CORE ,/ATE ,/&>DS
,9ITIATIVE1 ,APP5DIX ,A1 #BJAJ">4
  ,Z DESCRIB$ 9 ,*APT] #F1 BRACES >E
GR\P+ SYMBOLS4 ,"O ( ! USES ( BRACES 9 ?
*APT] IS TO REPRES5T SETS4 ,:5 A SET
9CLUDES A LI/ ( NUMB]S1 NO NUM]IC
9DICATORS >E US$ S9CE X IS AN 5CLOS$ LI/
"<,RULE #AA1 ^S#AJ">4
  ,! LEFT "<OP5"> BRACE "<DOTS #D-#F1
DOTS #A-#B-#C-#E-#F"> IS WRITT5 AT !
2G9N+ ( ! SET4
  .=.(
  ,! "R "<CLOSE"> BRACE "<DOTS #D-#F1
DOTS #B-#C-#D-#E-#F"> IS WRITT5 AT   #AJ
               ,*APT] #H           A#H4E
! 5D ( ! SET4
  .=.)

_% .(10, 11, 12.) _:
  ,R1D3 ! SET ) ELE;TS T51 ELEV51 &
    TWELVE

  ,"O V]TICAL B> C REPRES5T ! ^WS ^1S*
^1T4 ,9 ? CASE1 "! M/ 2 A SPACE 2F & AF
! V]TICAL B> 9 ,NEME? ,CODE "<,RULE
,,XX1 ^S#ADE">4 ,? V]TICAL B> IS USUALLY
US$ )9 SET NOTA;N TO REPRES5T DOMA9
"<POSSIBLE VALUES ( ;X"> OR RANGE
"<POSSIBLE VALUES ( ;Y">4 ,DOMA9 & RANGE
WRITT5 9 SET NOTA;N NORMALLY APPE> 9 A
SET ( BRACES4
------------------------------------#H4F

_% .(X \ X /.K #0.) _:
  ,R1D3 ,! SET ( ALL ;X S* T ;X IS N
    EQUAL TO #J4
_% .(Y \ -#3 "K: Y "K: #3.) _:
  ,R1D3 ,! SET ( ALL ;Y S* T -#C IS LESS
    ?AN OR EQUAL TO ;Y IS LESS ?AN   #AA
               ,*APT] #H           A#H4F
    OR EQUAL TO #C4

  ,(T5 A ^1VECTOR IS REPRES5T$ BY A
L[]CASE LR4 ,A VECTOR IS AN OBJECT T HAS
BO? A MAGNITUDE & A DIREC;N4 ,PICTURE A
VECTOR Z A DIRECT$ L9E SEG;T1 ^: L5G? IS
! MAGNITUDE ( ! VECTOR & ) AN >R[ ON "O
5D 9DICAT+ ! DIREC;N4 ,:5 ! "N ( !
VECTOR IS PLAC$ 2T TWO V]TICAL B>S1 !
EXPRES.N M1NS ! ^1MAGNITUDE ( ! VECTOR4

_% \U1\ .K #5 _:
  ,R1D3 ,! MAGNITUDE ( ;U SUB #A EQUALS
    #E4

  ,"S"TS ! MAGNITUDE ( A VECTOR IS
WRITT5 ) TWO V]TICAL B>S 2F & AF1 B X /
M1NS MAGNITUDE :E!R "! IS A SET ( S+LE
V]TICAL B>S OR A SET ( TWO V]TICAL B>S
2F & AF ! "N ( ! VECTOR4

_% \\V\\ .K >X^2"+Y^2"] _:
  ,R1D3 ,! MAGNITUDE ( ;V EQUALS ! SQU>E
    ROOT ( ;X SQU>$ PLUS ;Y SQU>$    #AB
               ,*APT] #H           B#H4F
    5D ROOT4

  ,A LR PLAC$ 2T V]TICAL B>S OR D\BLE
V]TICAL B>S C AL M1N ! ^1NORM ^1( "S?+
S* Z ! NORM ( FUNC;NS4 ,9 FACT1 A LR
PLAC$ 2T V]TICAL B>S 9 MA!MATICS HAS A
LOT ( DI6]5T 3TEXTS4 ,H["E1 9 ,NEME?
,CODE V]TICAL B>S >E ALW BRLD ! SAME WAY
Z PREVI\SLY DESCRIB$4 ,! NUMB] ( V]TICAL
B>S 9 BRL ALW FOLL[ ! NUMB] ( V]TICAL
B>S US$ 9 PR9T4
------------------------------------#H4G

_% \\G\\ _:
  ,R1D3 ! NORM ( ;G
_% \X\ _:
  ,R1D3 ! NORM ( ;X

  ,A CAPITAL LR IS (T5 US$ TO REPRES5T !
"N ( A MATRIX 9 ALGEBRA4 ,:5 V]TICAL B>S
>E PLAC$ >.D ? "N1 X M1NS ! ^7DET]M9ANT
( ! MATRIX^'4 ,LAT] 9 ! *APT]1 EXAMPLES
( MATRICES W 2 PROVID$4
                                     #AC
               ,*APT] #H           A#H4G
_% \,B\ .K -#4 _:
  ,R1D3 ,! DET]M9ANT ( MATRIX ;,B EQUALS
    NEGATIVE F\R4

  ,EXAMPLE #H4A 3TA9S ?REE EXAMPLES (
ABSOLUTE VALUE & ?REE EXAMPLES ( O!R
USES = ! V]TICAL B>4 ,NOTICE T 9 ! ?IRD
EXAMPLE ! OP5+ ,NEME? ,CODE 9DICATOR &
,NEME? ,CODE T]M9ATOR >E US$ 9 ! ^W
PROBLEM4

      ,EXAMPLE #H4A
_% \-5\ .K \5\ .K #5 _:
  ,R1D3 ,! ABSOLUTE VALUE ( NEGATIVE
    FIVE EQUALS ! ABSOLUTE VALUE ( FIVE
    EQUALS FIVE4
_% Y .K \8-X\ _:
  ,R1D3 ;Y EQUALS OP5 ABSOLUTE VALUE
    EI<T M9US ;X CLOSE ABSOLUTE VALUE4
------------------------------------#H4H
,SOLVE = ;X IF _% \X-2\ "K: #7 _:4
  ,R1D3 ,SOLVE = ;X IF OP5 ABSOLUTE
    VALUE ;X M9US TWO CLOSE ABSOLUTE
    VALUE IS LESS ?AN OR EQUAL TO    #AD
               ,*APT] #H           A#H4H
    SEV54
_% .(X \ X .1 #1.) _:
  ,R1D3 ,! SET ( ALL ;X S* T ;X IS GRT]
    ?AN "O4
_% \,D\ .K #35 _:
  ,R1D3 ,! DET]M9ANT ( ;,D EQUALS
    ?IRTY-FIVE4
_% \\F-H\\ _:
  ,R1D3 ! NORM ( ;F M9US ;H

    ,EXP&$ ,USE ( ,GR\P+ ,SYMBOLS
  ,Z DESCRIB$ 9 ,*APT] #F1 P>5!SES &
BRACKETS >E AL GR\P+ SYMBOLS4 ,!Y >E BO?
FREQU5TLY US$ 9 MA? EXPRES.NS4
  ,! LEFT "<OP5"> P>5!SIS IS WRITT5 DOTS
#A-#B-#C-#E-#F & ! "R "<CLOSE"> P>5!SIS
IS WRITT5 DOTS #B-#C-#D-#E-#F4
  _% ( ) _:
  ,! LEFT "<OP5"> BRACKET IS WRITT5 DOT
#D1 DOTS #A-#B-#C-#E-#F & ! "R "<CLOSE">
BRACKET IS WRITT5 DOT #D1 DOTS
#B-#C-#D-#E-#F4
  _% @( @) _:
  ,"H IS AN EXAMPLE TO REVIEW !      #AE
               ,*APT] #H           B#H4H
USE ( P>5!SES & BRACKETS COV]$ 9 PREVI\S
*APT]S4
------------------------------------#H4I

_% #2@(3-(5+4)@) _:
  ,R1D3 TWO OP5 BRACKET ?REE M9US OP5
    P>5!SIS FIVE PLUS F\R CLOSE P>5!SIS
    CLOSE BRACKET

  ,P>5!SES >E US$ 9 FUNC;N NOTA;N1 :
/UD5TS >E F/ 9TRODUC$ TO 9 ,ALGEBRA #A
"<,COMMON ,CORE ,/ATE ,/&>DS ,9ITIATIVE1
#BJAJ2 ,M>YL& ,/ATE ,DE"P;T ( ,$UCA;N1
#BJAE">4 ,A FUNC;N1 BY DEF9I;N1 HAS NO
TWO PO9TS ON ! GRAPH ) ! SAME
;X-COORD9ATE4
  ^1,FUNC;N ^1NOTA;N IS R1D 8;F ( ;X0 &
IS WRITT5 US+ A S+LE LR S* Z ;F FOLL[$
BY AN ;X 5CLOS$ 9 P>5!SES4 ,NOTICE T EV5
?\< X IS R1D 8;F ( ;X10 ! ^W 8(0 IS N
WRITT5 :5 US+ FUNC;N NOTA;N4 ,! FUNC;N
DOESN'T ALW H TO 2 "ND ;F1 B FUNC;NS >E
COMMONLY "ND US+ ! LRS ;F1 ;G1 OR ;H4
  _% F(X) _:                         #AF
               ,*APT] #H           A#H4I
  ,FUNC;NS AL H A DOMA9 & RANGE4 ,!
DOMA9 REF]S TO ! SET ( 9PUTS OR ;X
VALUES & ! RANGE REF]S TO ! SET ( \TPUTS
OR ;Y VALUES4 ,! DOMA9 & RANGE >E (T5
WRITT5 9 ^19T]VAL ^1NOTA;N4 ,/UD5TS >E
9TRODUC$ TO 9T]VAL NOTA;N 9 ,ALGEBRA #A
& ,ALGEBRA #B "<,COMMON ,CORE ,/ATE
,/&>DS ,9ITIATIVE1 ,APP5DIX ,A1 #BJAJ">4
  ,9T]VAL NOTA;N USES GR\P+ SYMBOLS S* Z
P>5!SES OR BRACKETS4 ,! NUMB]S AT !
2G9N+ & ! 5D ( ! 9T]VAL >E WRITT5 9SIDE
! GR\P+ SYMBOLS & >E SEP>AT$ BY A COMMA
FOLL[$ BY A SPACE4
  ,:5 WRIT+ 9T]VAL NOTA;N1 A SET (
P>5!SES1 A SET ( BRACKETS1 OR "O ( EA*
CD 2 US$4 ,! P>5!SIS M1NS T ! NUMB] NEXT
TO ! P>5!SIS _1IS _1N 9CLUD$ 9 ! 9T]VAL4
,AN OP5 9T]VAL IS WRITT5 ) BO? ! OP5+ &
CLOS+ P>5!SES4 ,! BRACKET M1NS T ! NUMB]
NEXT TO ! BRACKET _1IS 9CLUD$ 9 !
9T]VAL4 ,A CLOS$ 9T]VAL IS WRITT5 ) BO?
! OP5+ & CLOS+ BRACKETS4
  ,ANO!R SYMBOL (T5 US$ 9 9T]VAL NOTA;N
IS ! ^19F9;Y ^1SYMBOL4 ,X IS US$     #AG
               ,*APT] #H           B#H4I
TO REPRES5T T ! 9T]VAL 3T9UES ="E4 ,X
TAKES TWO CELLS TO WRITE ! 9F9;Y SYMBOL
9 ,NEME? ,CODE4 ,! 9F9;Y SYMBOL IS DOT
#F1 FOLL[$ BY DOTS #A-#B-#C-#D-#E-#F4
-----------------------------------#H4AJ
  .=,=
  ,EXAMPLE #H4B 3TA9S FIVE EXAMPLES4 ,!
F/ TWO EXAMPLES %[ FUNC;NS1 & ! LA/ ?REE
EXAMPLES %[ 9T]VAL NOTA;N4 ,! LA/ ?REE
EXAMPLES USE P>5!SES1 BRACKETS1 OR A
COMB9A;N ( P>5!SES & BRACKETS TO WRITE
9T]VAL NOTA;N4

      ,EXAMPLE #H4B
F(X) .K -#3(X+2)-4
  ,R1D3 ;F ( ;X EQUALS NEGATIVE ?REE OP5
    ABSOLUTE VALUE ;X PLUS TWO CLOSE
    ABSOLUTE VALUE M9US F\R4
G(X) .K #7\X-5\+1
  ,R1D3 ;G ( ;X EQUALS SEV5 OP5 ABSOLUTE
    VALUE ;X M9US FIVE CLOSE ABSOLUTE
    VALUE PLUS "O4
(-6, 9)
  ,R1D3 ! OP5 9T]VAL F -#F TO #I     #AH
               ,*APT] #H          A#H4AJ
@(0, 8@)
  ,R1D3 ! CLOS$ 9T]VAL F #J TO #H
(-,=, 12@)
  ,R1D3 ! 9T]VAL F NEGATIVE 9F9;Y TO
    #AB1 9CLUD+ #AB

    ,HOLL[ ,DOT
  ,! HOLL[ DOT IS US$ = DEGREES &
COMPOSITE FUNC;NS4 ,/UD5TS >E F/
9TRODUC$ TO ! HOLL[ DOT US$ 9 DEGREES 9
ELE;T>Y S*OOL &
-----------------------------------#H4AA
!N COMPOSITE FUNC;NS 9 ,ALGEBRA #A OR #B
"<,COMMON ,CORE ,/ATE ,/&>DS ,9ITIATIVE1
,APP5DIX ,A1 #BJAJ">4
  ,REMEMB] T ! DEGREE SYMBOL IS ?REE
CELLS L;G & 9CLUDES ! SUP]SCRIPT
9DICATOR "<DOTS #D-#E"> FOLL[$ BY !
HOLL[ DOT "<DOTS #D-#F1 DOTS #A-#F">4
  ,! ^1HOLL[ ^1DOT XF IS TWO CELLS L;G &
IS WRITT5 DOTS #D-#F1 DOTS #A-#F4
  .=.*
  ,! HOLL[ DOT IS AL US$ :5 REPRES5T+
COMPOSITE FUNC;NS4                   #AI
               ,*APT] #H          A#H4AA
  ,9 ALGEBRA1 A ^1COMPOSITE ^1FUNC;N IS
A FUNC;N T DEP5DS ON ANO!R FUNC;N4 ,X IS
CR1T$ :5 "O FUNC;N IS SUB/ITUT$ 9TO
ANO!R FUNC;N4
  ,:5 R1D+ A FUNC;N1 SAY 8(0 :5 Y R1* !
HOLL[ DOT4 ,TO R1D _% F.*G _:1 SAY 8;F (
;G40 ,? M1NS T _% G(X) _: WD 2 SUB/ITUT$
9TO _% F(X) _: :5 CR1T+ ! COMPOSITE
FUNC;N4
  _% F.*G _:

    ,ANGLE ,BRACKETS
  ,ANGLE BRACKETS >E F/ 9TRODUC$ :5
/UD5TS />T "W+ ) VECTORS 9 PRECALCULUS4
,ANGLE BRACKETS >E ANO!R TYPE ( GR\P+
SYMBOL "<,COMMON ,CORE ,/ATE ,/&>DS
,9ITIATIVE1 ,APP5DIX ,A1 #BJAJ">4
  ,X TAKES ?REE CELLS TO WRITE ! ANGLE
BRACKETS4
  ^7OP5+ "<LEFT"> ANGLE BRACKET^' "<DOTS
#D-#F1 DOTS #D-#F1 DOTS #A-#B-#C-#E-#F">
  ^7CLOS+ "<"R"> ANGLE BRACKET^' "<DOTS
#D-#F1 DOTS #D-#F1 DOTS #B-#C-#D-#E-#F">
  _% ..( ..) _:                      #BJ
               ,*APT] #H          B#H4AA
  ,! NUM]IC 9DICATOR IS N USUALLY US$ :5
NUMB]S >E WRITT5 9SIDE ( ! ANGLE
BRACKETS 2C ! NUMB]S >E 3SID]$ "P ( AN
5CLOS$ LI/4
-----------------------------------#H4AB
  ,EXAMPLE #H4C 3TA9S FIVE PROBLEMS4 ,!
F/ EXAMPLE %[S A HOLL[ DOT US$ TO
REPRES5T DEGREES4 ,! SECOND EXAMPLE %[S
! HOLL[ DOT US$ 9 COMPOSITE FUNC;NS4 ,!
?IRD EXAMPLE %[S T "S"TS "! C 2 A NUMB]
9/1D ( AN ;X 9 ! P>5!SES4 ,! LA/ TWO
EXAMPLES %[ ! USE ( ANGLE BRACKETS4

      ,EXAMPLE #H4C
_% #135^.* _:
  ,R1D3 "O HUNDR$ ?IRTY-FIVE DEGREES
_% (H.*G)(X) .K H(G(X)) _:
  ,R1D3 ;H ( ;G ( ;X EQUALS ;H ( ;G (
    ;X4
_% F(2) .K -#4(2)+3 _:
  ,R1D3 ;F ( TWO EQUALS NEGATIVE F\R OP5
    P>5!SIS TWO CLOSE P>5!SIS PLUS ?REE4
_% ..(-7, 60^.*"..) _:
  ,R1D3 OP5 ANGLE BRACKET NEGATIVE   #BA
               ,*APT] #H          A#H4AB
    SEV5 COMMA SIXTY DEGREES CLOSE ANGLE
    BRACKET
_% ..(?1/2#, ?>3]/2#..) _:
  ,R1D3 OP5 ANGLE BRACKET "O-HALF COMMA
    SQU>E ROOT ( ?REE 5D ROOT OV] TWO
    CLOSE ANGLE BRACKET
-----------------------------------#H4AC

      ,PRACTICE #H4A
    ,9T]L9E ! FOLL[+ PROBLEMS T 3TA9
  GR\P+ SYMBOLS & HOLL[ DOTS4
#1_4 (G.*F)(6) .K \6\+2
#2_4 @(-8, ,=)
#3_4 .(Y \ Y "K #9.)
#4_4 ..(7, 30^.*"..)
#5_4 \,M\ .K -#10

    ,USE YR BRLWRIT] TO WRITE ! PROBLEMS
  T 3TA9 GR\P+ SYMBOLS & HOLL[ DOTS4
      @.<,MA? 3T5T HAS BE5 SPELL$ \ TO
    ALL[ = PRACTICE4@.>
#A4 ,OP5-BRACE ;X V]TICAL-B> ;X IS GRT]
  ?AN OR EQUAL TO Z]O CLOSE-BRACE
#B4 ,V]TICAL-B> ;,C V]TICAL-B>       #BB
               ,*APT] #H          A#H4AC
  EQUALS AD M9US BC
#C4 ,OP5-P>5!SIS ;H HOLL[-DOT ;F
  HOLL[-DOT ;G CLOSE-P>5!SIS OP5-P>5!SIS
  ;X CLOSE-P>5!SIS EQUALS ! SQU>E ROOT (
  ;X M9US TWO @.<5D ROOT@.> PLUS "O
#D4 ,OP5-P>5!SIS NEGATIVE TWO1 N9E
  CLOSE-BRACKET
#E4 ,OP5 ANGLE-BRACKET ?REE1 N9ETY
  DEGREES CLOSE ANGLE-BRACKET

    ,5L>G$ ,GR\P+ ,SYMBOLS
  ,/UD5TS >E 9TRODUC$ TO 5L>G$ GR\P+
SYMBOLS :5 !Y />T "W+ ) SY/EMS ( EQUA;NS
& PIECEWISE-DEF9$ FUNC;NS 9 ,ALGEBRA #A
& MATRICES 9 PRECALCULUS "<,COMMON ,CORE
,/ATE ,/&>DS ,9ITIATIVE1 ,APP5DIX ,A1
#BJAJ">4
  ,:5 5L>G+ A GR\P+ SYMBOL S* Z A
P>5!SIS1 BRACKET1 BRACE1 OR V]TICAL B>1
A DOT #F IS US$ 2F ! LA/ CELL ( ! GR\P+
SYMBOL J L A DOT #F IS US$ 9 FRONT ( A
LR TO CAPITALIZE X4 ,NOTICE ! PLACE;T (
! DOT #F 9 ! LI/ 2L4
                                     #BC
               ,*APT] #H          B#H4AC

.=,( .<DOT #F1 DOTS #A-#B-#C-#E-#F.>
  ,5L>G$ OP5+ "<LEFT"> P>5!SIS
.=,) .<DOT #F1 DOTS #B-#C-#D-#E-#F.>
  ,5L>G$ CLOS+ "<"R"> P>5!SIS
-----------------------------------#H4AD
.=@,( .<DOT #D1 DOT #F1 DOTS
  #A-#B-#C-#E-#F.> ,5L>G$ OP5+ "<LEFT">
  BRACKET
.=@,) .<D #D1 DOT #F1 DOTS
  #B-#C-#D-#E-#F.> ,5L>G$ CLOS+ "<"R">
  BRACKET
.=.,( .<DOTS #D-#F1 DOT #F1 DOTS
  #A-#B-#C-#E-#F.> ,5L>G$ OP5+ "<LEFT">
  BRACE
.=.,) .<DOTS #D-#F1 DOT #F1 DOTS
  #B-#C-#D-#E-#F.> ,5L>G$ CLOS+ "<"R">
  BRACE
.=,\ .<DOT #F1 DOTS #A-#B-#E-#F.> ,5L>G$
  V]TICAL B>

  ,! 5L>G$ GR\P+ SYMBOL IS US$ 9 BRL :5
MA!MATICAL EXPRES.NS >E >RANG$ ON TWO OR
M L9ES 9 PR9T & A SIGN ( GR\P+ IS    #BD
               ,*APT] #H          A#H4AD
US$ TO UNIFY ! L9ES "<,RULE ,,XVIII
^S#ABF">4 ,! 5L>G$ GR\P+ SYMBOLS >E (T5
US$ = PIECEWISE-DEF9$ FUNC;NS1 SY/EMS (
EQUA;NS1 MATRICES1 & DET]M9ANTS4
  ^1,MA? ^1,REVIEW3

_4 ,PIECEWISE-DEF9$ FUNC;NS ,- A GR\P (
  TWO OR M FUNC;NS1 EA* HAV+ _! [N
  9T]VAL OR POR;N ( A S+LE FUNC;N4 ,X IS
  A FUNC;N BUILT F PIECES ( DI6]5T
  FUNC;NS OV] DI6]5T 9T]VALS4
_4 ,SY/EMS ( EQUA;NS ,- A GR\P ( TWO OR
  M EQUA;NS T >E SOLV$ TGR TO F9D !
  9T]SEC;N ( _! GRAPHS1 IF "O EXI/S4
_4 ,MATRICES ,- RECTANGUL> >RAY OR TABLE
  ( NUMB]S1 SYMBOLS1 OR EXPRES.NS1
  >RANG$ 9 R[S & COLUMNS
_4 ,DET]M9ANTS ,- SPECIAL NUMB]S T C 2
  CALCULAT$ F SQU>E MATRICES "<MATRICES
  ) ! SAME NUMB] ( R[S Z COLUMNS">

  ^7,KEEP ^! PO9TS 9 M9D :5 US+ 5L>G$
GR\P+ SYMBOLS3^'
                                     #BE
               ,*APT] #H          B#H4AD

_4 ,>RANGE;TS ) 5L>G$ GR\P+ SYMBOLS >E
    3SID]$ SPATIAL PROBLEMS S L1VE A
    BLANK L9E ABV & 2L ! >RANGE;T
    "<,RULE ,,XXV ^S#AHE;A">4
_4 ,"! M/ 2 AT L1/ "O 5L>G$ GR\P+ SYMBOL
    ON EA* L9E ( ! >RANGE;T "<,RULE
    ,,XVIII ^S#ABF">4
_4 ,! 5L>G$ GR\P+ SYMBOLS M/ 2 L9$ UP
    V]TICALLY "<,RULE ,,XVIII ^S#ABF">4
_4 ,FOLL[ PR9T REG>D+ ! PRES;E OR ABS;E
    ( 5L>G$ GR\P+ SYMBOLS "<,RULE
    ,,XVIII ^S#ABF">4
_4 ,"S"TS AN >RANGE;T US+ 5L>G$ GR\P+
    SYMBOLS H PROBLEM ID5TIFI]S1 "PS (
    AN EXPRES.N1 PUNCTUA;N1 ETC4 3NECT$
    TO ! >RANGE;T4 ,^! M/ 2 PLAC$ ON !
    TOP L9E ( ! >RANGE;T 9 BRL REG>D.S (
    XS LOCA;N 9 PR9T "<,RULE ,,XXV
    ^S#AHE;B IV">4
-----------------------------------#H4AE
_4 ,= SY/EMS ( EQUA;NS1 :5 ELE;TS ( !
    SY/EM >E V]TICALLY ALIGN$ 9 PR9T1 !
    ALIGN;T IS RETA9$ 9 BRL          #BF
               ,*APT] #H          A#H4AE
    "<,RULE ,,XVIII ^S#ABH">4
_4 ,= MATRICES & DET]M9ANTS3
  _4 ,EA* 5TRY M/ 2 MOV$ Z F> LEFT Z
    POSSIBLE 9 XS COLUMN "<,RULE ,,XXIV
    ^S#AHC;A">4
  _4 ,"O COLUMN ( BLANK CELLS M/ 2 LEFT
    2T ! COLUMNS ( ! >RANGE;T "<,RULE
    ,,XXIV ^S#AHC;A">4
  _4 ,! NUM]IC 9DICATOR M/ 2 US$ )
    NUM]IC 5TRIES "<,RULE ,,II ^S#I;C">4

  ,KEEP ^! RULES 9 M9D :5 LOOK+ AT !
FOLL[+ EXAMPLE4 ,9 "PICUL>1 NOTICE "! IS
A BLANK L9E AF ! DIREC;NS & 2F ! F/
SPATIAL PROBLEM 9 ! EXAMPLE 2L4

      ,EXAMPLE #H4D
    ,SOLVE ! FOLL[+ PROBLEMS4 _%

#1_4 Y .K .,(-X-2, X "K: -#2
          .,(>4-X^2"], -#2 "K X "K #2
          .,(X-2, X .1: #2

                                     #BG
               ,*APT] #H          B#H4AE

#2_4 @,(#1 #1 #1  \ #2@,)
     @,(#0 #1 -#3 \ #1@,)
     @,(#2 #1 #5  \ #0@,)

_:

-----------------------------------#H4AF
  ,9 ,PROBLEM #A ABV1 NOTICE ! PROBLEM
NUMB] & ! 8;Y "70 IS L9$ UP ) ! F/ L9E 9
BRL EV5 ?\< IS X L9$ UP ) ! SECOND L9E 9
PR9T4
  ,9 ,PROBLEM #B ABV1 : IS CALL$ AN
^1AUG;T$ ^1MATRIX1 NOTICE ! FOLL[+3

_4 ,! V]TICAL B> IS N 3SID]$ 5L>G$ 2C X
  IS ALR 9 5L>G$ BRACKETS
_4 ,! NUM]IC 9DICATORS >E US$
_4 ,EA* 5TRY IS MOV$ Z F> LEFT Z
  POSSIBLE 9 XS COLUMN
_4 ,"O COLUMN ( BLANK CELLS IS LEFT 2T !
  COLUMNS ( ! >RANGE;T4

  ,EXAMPLE #H4E 3TA9S FIVE           #BH
               ,*APT] #H          A#H4AF
EXAMPLES ( PIECEWISE-DEF9$ FUNC;NS1
SY/EMS ( EQUA;NS1 MATRICES1 &
DET]M9ANTS4 ,! GR\P+ SYMBOLS >E L9$ UP
V]TICALLY 9 ALL PROBLEMS4
  ,9 ! F/ EXAMPLE1 NOTICE ! V>IABLES A1
;B1 & ;C >E L9$ UP 9 BO? PR9T & BRL4
  ,9 ! SECOND EXAMPLE1 NOTICE ! PLUS
SIGN IS ON ! F/ L9E 9 BRL EV5 ?\< X IS
C5T]$ 9 PR9T4
  ,9 ! ?IRD EXAMPLE1 NOTICE ! NUM]IC
9DICATORS >E US$1 EA* 5TRY IS MOV$ Z F>
LEFT Z POSSIBLE 9 XS COLUMN1 & "O COLUMN
( BLANK CELLS IS LEFT 2T ! COLUMNS ( !
>RANGE;T4 ,AL1 NOTICE T ! ^WS 2F !
DET]M9ANT & ! "Q M>K AF ! DET]M9ANT >E
ON ! F/ L9E 9 BRL EV5 ?\< !Y >E C5T]$ 9
PR9T4
  ,ALL ( ! FOLL[+ EXAMPLES CD 2 R1D _M
DI6]5T WAYS1 S 3SULT YR MA? T1*] TO
DET]M9E H[ !Y WD L ! PROBLEMS R1D4



                                     #BI
               ,*APT] #H          B#H4AF

      ,EXAMPLE #H4E _%

#1_4 ,( A+B+ C .K #2
     ,(   B-3C .K #1
     ,(2A+B+5C .K #0

-----------------------------------#H4AG

#2_4 @,(-#7  #5@,)+@,(-#4 #0 @,)
     @,(-#10 #3@,) @,(#6  -#2@,)

#3_4 ,:AT IS ,\#-6 #4 ,\_8 _:
             ,\#9  -#5,\

    ,NON-,USE ( ,5L>G$ ,GR\P+ ,SYMBOLS
  ,D N USE 5L>G$ GR\P+ SYMBOLS 9 !
FOLL[+ SITUA;NS "<,RULE ,,XVIII
^S#ABG">3

_4 ,TO COV] A FRAC;N
  _% (?2X+4/3#)(?3X-6/2#) _:
    ,R1D3 OP5 P>5!SIS OP5 FRAC;N TWO ;X
      PLUS F\R OV] ?REE CLOSE        #CJ
               ,*APT] #H          A#H4AG
      FRAC;N CLOSE P>5!SIS OP5 P>5!SIS
      OP5 FRAC;N ?REE ;X M9US SIX OV]
      TWO CLOSE FRAC;N CLOSE P>5!SIS
_4 ,TO WRITE A B9OMIAL COE6ICI5T
  ,! B9OMIAL COE6ICI5T USES REGUL>
      P>5!SES4 ,X AL USES ! DIRECTLY "U
      9DICATOR "<DOTS #A-#D-#F"> 2T !
      TOP NUMB] & ! BOTTOM NUMB]4 ,= M
      9=MA;N1 SEE ,RULE ,,XIV ^S#IJ4
  _% (5%4) _:
    ,R1D3 FIVE *OOSE F\R
-----------------------------------#H4AH
_4 ,TO WRITE MAT]IAL O3UPY+ A L>GE AM.T
      ( V]TICAL SPACE S* Z A SQU>E ROOT
      9SIDE ( ANO!R SQU>E ROOT
  ,! F/ 9N] RADICAL ORD]-(-RADICAL
      9DICATOR "<DOTS #D-#F"> IS PLAC$ 9
      FRONT ( ! 9N] RADICAL "<DOTS
      #C-#D-#E"> & ! 9N] T]M9A;N
      9DICATOR "<DOTS #A-#B-#D-#E-#F">4
      ,= M 9=MA;N1 SEE ,RULE ,,XV
      ^S#AJE4
  _% >.>125.]] _:
    ,R1D3 ! SQU>E ROOT ( ! SQU>E     #CA
               ,*APT] #H          A#H4AH
      ROOT ( "O HUNDR$ TW5TY-FIVE

      ,PRACTICE #H4B
    ,9T]L9E ! FOLL[+ PROBLEMS T 3TA9
  5L>G$ GR\P+ SYMBOLS4
_%
#1_4 (6%2)

#2_4 G(X) .K @,(2X-1, ,'= X "K #4
             @,(5-X, ,'= X .1: #4

#3_4 .,( 2X-3Y .K #6
     .,(-3X+4Y .K #12

#4_4 ,C .K @,(#7  -#1 #0 @,)
           @,(-#3 #8  -#9@,)

#5_4 ,\-#5  #6 ,\
     ,\-#10 #12,\

_:


                                     #CB
               ,*APT] #H           #H4AI

    ,USE YR BRLWRIT] TO WRITE ! PROBLEMS
  T 3TA9 5L>G$ GR\P+ SYMBOLS4
      @.<,MA? 3T5T HAS BE5 SPELL$ \ TO
    ALL[ = PRACTICE4 ,TRANSCRIB]S NOTES
    >E 9CLUD$ )9 PROBLEMS TO 9DICATE
    BR1KS 2T L9ES4 ,5L>G$ GR\P+ SYMBOLS
    APPLY TO ALL L9ES4@.>
#A4 ;Y EQUALS 5L>G$ OP5-BRACE @.<L9E
  #A@.> NEGATIVE Z]O PO9T FIVE ;X PLUS
  ?REE1 ;X IS LESS ?AN OR EQUAL TO
  NEGATIVE TWO @.<L9E #B@.> SEV5 PLUS
  @.<FRAC;N />T@.> ?REE OV] F\R
  @.<FRAC;N 5D@.> ;X1 ;X IS GRT] ?AN
  NEGATIVE TWO
#B4 ,5L>G$ OP5-P>5!SIS @.<L9E #A@.>
  NEGATIVE A PLUS FIVE ;B EQUALS T5
  @.<L9E #B@.> SEV5 ;B EQUALS TW5TY-"O
  @.<T]MS >E ALIGN$ 9 PR9T@.>
#C4 ;,D EQUALS 5L>G$ OP5-BRACKET @.<L9E
  #A@.> Z]O NEGATIVE TWO @.<L9E #B@.>
  SIX T5 @.<L9E #C@.> NEGATIVE EI<T "O
  5L>G$ CLOSE-BRACKET
#D4 ,5L>G$ V]TICAL-B> @.<L9E #A@.>   #CC
               ,*APT] #H          A#H4AI
  A ;B @.<L9E #B@.> ;C ;D 5L>G$
  V]TICAL-B>
#E4 ,OP5-P>5!SIS @.<FRAC;N />T@.> F\R
  OV] ;X PLUS ;Y @.<FRAC;N 5D@.>
  CLOSE-P>5!SIS M9US OP5-P>5!SIS
  @.<FRAC;N />T@.> TWO OV] ;X M9US ;Y
  @.<FRAC;N 5D@.> CLOSE P>5!SIS
      @.<,9 ITEM #E1 ! P>5!SIS APPE>
    L>G] 9 PR9T 2C !Y >E 5CLOS+
    FRAC;NS4@.>

    ,SET ,!ORY
  ,/UD5TS >E 9TRODUC$ TO SET !ORY 9
GEOMETRY "<,COMMON ,CORE ,/ATE ,/&>DS
,9ITIATIVE1 ,APP5DIX ,A1 #BJAJ">4
  ,"O ( ! SYMBOLS US$ 9 SET !ORY IS !
^1EMPTY ^1SET1 : IS AL REF]R$ TO Z !
^1NULL ^1SET4 ,AN EMPTY SET IS A SET )
NO ELE;TS 9 X4 ,"! >E TWO WAYS TO
REPRES5T ! EMPTY SET4 ,"O WAY IS TO USE
A SET ( BRACES ) A SPACE 2T !M4
  _% .( .) _:
  ,ANO!R WAY TO WRITE ! EMPTY SET IS TO
USE ! ^7EMPTY SET SYMBOL^'4 ,X       #CD
               ,*APT] #H          B#H4AI
TAKES TWO CELLS TO WRITE ! EMPTY SET
SYMBOL 9 ,NEME? ,CODE4 ,! EMPTY SET
SYMBOL IS WRITT5 DOTS #D-#E-#F1 DOTS
#C-#E-#F4
  .=_0
-----------------------------------#H4BJ
  ,TWO COMMON SIGNS ( OP]A;N US$ 9 SET
!ORY >E ! UNION & 9T]SEC;N SYMBOLS4
,S9CE ^! SYMBOLS >E 3SID]$ SIGNS (
OP]A;N1 "! >E NO SPACES ON EI SIDE ( !
SYMBOLS4
  ,X TAKES TWO CELLS TO WRITE ! UNION
SYMBOL 9 ,NEME? ,CODE4 ,! ^1UNION
^1SYMBOL IS WRITT5 DOTS #D-#F1 DOTS
#C-#D-#F4
  .=.+
  ,F9D+ ! UNION ( TWO SETS1 IS L UNIT+
!M OR PUTT+ !M TGR S T ALL ( ! COMB9$
ELE;TS >E 9 "O SET4

  ^1,T1*+ ^1,TIP3 ,! PLUS SIGN IS US$ 9
  ! UNION SYMBOL4 ,PO9T \ TO /UD5TS T !
  UNION SYMBOL IS J L 8ADD+0 ! TWO SETS
  TGR4 ,ANO!R WAY TO ?9K AB X IS T   #CE
               ,*APT] #H          A#H4BJ
  X LOOKS L A 8;,U0 OR ! PR9T UNION
  SYMBOL IF Y 3NECT ! DOTS4

  ,X TAKES TWO CELLS TO WRITE ! 9T]SEC;N
SYMBOL 9 ,NEME? ,CODE4 ,! ^19T]SEC;N
^1SYMBOL IS WRITT5 DOTS #D-#F1 DOTS
#A-#D-#F4
  .=.%
  ,F9D+ ! 9T]SEC;N ( TWO SETS IS L F9D+
! 9T]SEC;N OR OV]LAP 2T ! TWO SETS4 ,!
9T]SEC;N IS ! SET ( ELE;TS T >E ! SAME 9
BO? ( ! ORIG9AL SETS4

  ^1,T1*+ ^1,TIP3 ,S9CE F9D+ AN 9T]SEC;N
  ( TWO SETS IS ! SAME Z F9D+ ! ELE;TS T
  >E ! SAME OR %>$ 2T TWO SETS1 PO9T \
  TO /UD5TS T DOTS #A-#D-#F 9 ! 9T]SEC;N
  SYMBOL IS L ! 8SH0 3TRAC;N US$ AT !
  2G9N+ ( ! ^W %>$4 ,ANO!R WAY TO ?9K AB
  X IS T X LOOKS L ! PR9T 9T]SEC;N
  SYMBOL IF Y 3NECT ! DOTS4

  ,TWO COMMON SIGNS ( COMP>ISON US$ 9
SET !ORY >E ! 8IS A SUBSET (0 &      #CF
               ,*APT] #H          B#H4BJ
8IS AN ELE;T (0 SYMBOLS4 ,^! SYMBOLS >E
SIGNS ( COMP>ISON1 S "! IS A SPACE 2F &
AF ! SYMBOLS4
  ,X TAKES F\R CELLS TO WRITE ! 8^7IS A
SUBSET (0^' SYMBOL 9 ,NEME? ,CODE4 ,X IS
WRITT5 DOTS #D-#E-#F1 DOT #E1 DOTS
#D-#F1 DOTS #A-#E-#F4
-----------------------------------#H4BA
  .=_"K:
  ,A SUBSET IS A SMALL] "P ( A BI7] SET4

  ^1,T1*+ ^1,TIP3 ,PO9T \ TO /UD5TS T !
  BRL 3FIGURA;N = ! 8IS A SUBSET (0
  SYMBOL 9CLUDES A LESS ?AN OR EQUAL TO
  SIGN4

  ,X TAKES TWO CELLS TO WRITE ! ^78IS AN
ELE;T (0^' SYMBOL ,9 ,NEME? ,CODE4 ,X IS
WRITT5 DOT #E1 DOTS #A-#E4
  .=@E
  ,AN ELE;T IS AN OBJECT OR MEMB] ( A
SET4 ,CAPITAL LRS C 2 US$ TO REPRES5T !
"N ( A "PICUL> SET4
                                     #CG
               ,*APT] #H          A#H4BA

  ^1,T1*+ ^1,TIP3 ,PO9T \ TO /UD5TS T !
  LR 8;E0 = 8ELE;T0 IS 9 ! BRL 3FIGURA;N
  = ! 8IS AN ELE;T (0 SYMBOL4

  ,EXAMPLE #H4F 3TA9S SEV5 EXAMPLES US+
SET !ORY NOTA;N4 ,9 ! F/ F\R EXAMPLES1
"! IS NO SPACE >.D ! UNION OR 9T]SEC;N
SYMBOLS 2C !Y >E SIGNS ( OP]A;N4 ,9 !
LA/ ?REE EXAMPLES1 "! IS A SPACE 2F & AF
! SYMBOLS = 8IS A SUBSET (0 & 8IS AN
ELE;T (0 2C !Y >E SIGNS ( COMP>ISON4

      ,EXAMPLE #H4F
_% ,A.+,B _:
  ,R1D3 ,A UNION ;,B
_% .(1, 3.).+.(2, 4.) .K .(1, 2, 3, 4.)
    _:
  ,R1D3 ,! SET ) ELE;TS "O & ?REE UNION
    ! SET ) ELE;TS TWO & F\R EQUALS !
    SET ) ELE;TS "O1 TWO1 ?REE1 & F\R4
-----------------------------------#H4BB
_% .(1, 3.).%.(2, 3.) .K .(3.) _:
  ,R1D3 ,! SET ) ELE;TS "O & ?REE    #CH
               ,*APT] #H          A#H4BB
    9T]SEC;N ! SET ) ELE;TS TWO & ?REE
    EQUALS ! SET ) ELE;T ?REE4
_% .(1, 3.).%.(2, 4.) .K _0 _:
  ,R1D3 ,! SET ) ELE;TS "O & ?REE
    9T]SEC;N ! SET ) ELE;TS TWO & F\R
    EQUALS ! EMPTY SET4
_% .(2, 4.) _"K: .(1, 2, 3, 4.) _:
  ,R1D3 ,! SET ) ELE;TS TWO & F\R IS A
    SUBSET ( ! SET ) ELE;TS "O1 TWO1
    ?REE1 & F\R4
_% .( .) _"K: .(,A, ,B, ,C, ,D.) _:
  ,R1D3 ,! EMPTY SET IS A SUBSET ( ! SET
    ) ELE;TS ,A1 ;,B1 ;,C1 & ;,D4
_% #4 @E .(2, 4, 6, 8.) _:
  ,R1D3 ,F\R IS AN ELE;T ( ! SET )
    ELE;TS TWO1 F\R1 SIX1 & EI<T4

      ,PRACTICE #H4C
    ,9T]L9E ! FOLL[+ PROBLEMS T 3TA9
  SYMBOLS 9 SET !ORY4 _%
#1_4 .(S, T.) _"K: .(S, E, T.)
#2_4 ,Y.+_0 .K ,Y
#3_4 #16 @E .(1, 4, 9, 16, 25.)
                                     #CI
               ,*APT] #H           #H4BC
#4_4 .(PIG, COW.).%.(CAT, DOG.) .K .( .)
#5_4 ,A.+,B .K .(1, 3, 5, 15.) _:

    ,USE YR BRLWRIT] TO WRITE ! FOLL[+
  PROBLEMS T 3TA9 SYMBOLS 9 SET !ORY4
      @.<,MA? 3T5T HAS BE5 SPELL$ \ TO
    ALL[ = PRACTICE4@.>
#A4 ,OP5-BRACE T51 FIFTE51 TW5TY
  CLOSE-BRACE 9T]SEC;N OP5-BRACE T51
  TW5TY1 ?IRTY CLOSE-BRACE EQUALS
  OP5-BRACE T51 TW5TY CLOSE-BRACE
#B4 ;,D IS A SUBSET ( ,A
#C4 ,OP5-BRACE A1 ;B1 ;C CLOSE-BRACE
  UNION OP5-BRACE ;X1 ;Y1 ;Z CLOSE-BRACE
  EQUALS OP5-BRACE A1 ;B1 ;C1 ;X1 ;Y1 ;Z
  CLOSE-BRACE
#D4 A IS AN ELE;T ( OP5-BRACE ;M1 A1 ;T1
  ;H CLOSE-BRACE
#E4 ;,E 9T]SEC;N ;,F EQUALS EMPTY-SET

    ,REVIEW ( ,SUP]SCRIPTS & ! ,BASEL9E
    ,9DICATOR
  ,Z 9TRODUC$ 9 ,*APT] #F1 !
^1SUP]SCRIPT ^19DICATOR "<DOTS       #DJ
               ,*APT] #H          A#H4BC
#D-#E"> IS US$ TO REPRES5T EXPON5TS1
P[]S1 OR RAIS$ "*S S* Z ! DEGREE SIGN OR
9V]SE4 ,X IS US$ AF ! BASE & 2F ! ACTUAL
RAIS$ "* OR EXPON5T4 ,"! IS N A
CORRESPOND+ SIGN 9 PR9T = ! SUP]SCRIPT
9DICATOR4 ,IF A MA!MATICAL SYMBOL1 S* Z
A SIGN ( OP]A;N1 GR\P+ SYMBOL1 T]M9A;N
9DICATOR1 OR FRAC;N L9E1 COMES AF !
EXPON5T1 A ^1BASEL9E ^19DICATOR "<DOT
#E"> IS US$ AF ! EXPON5T TO 9DICATE !
EXPON5T HAS 5D$ & "! IS A RETURN TO
BASEL9E4
  ,NO BASEL9E 9DICATOR IS NE$$ IF A
SPACE OR O!R LEVEL 9DICATOR FOLL[S A
SUP]SCRIPT OR SUBSCRIPT4 ,"! IS NO PR9T
EQUIVAL5T = ! BASEL9E 9DICATOR4
  ,A COMMON =MULA T USES SUP]SCRIPT &
BASEL9E 9DICATORS IS ! ,PY?AGOR1N ,!OREM
= A "R TRIANGLE ) SIDES A1 ;B1 & ;C4 ,!
HYPOT5USE ;C IS ! L;GE/ SIDE & OPPOSITE
! "R ANGLE4

_% A^2"+B^2 .K C^2 _:
  ,R1D3 A SQU>$ PLUS ;B SQU>$        #DA
               ,*APT] #H          B#H4BC
    EQUALS ;C SQU>$4

-----------------------------------#H4BD
  ,! SUP]SCRIPT 9DICATOR IS NE$$ 2F EA*
EXPON5T4 ,FOLL[+ ! 8A SQU>$10 ! BASEL9E
9DICATOR IS NE$$ 2C ! 8A SQU>$0 IS
FOLL[$ BY ! PLUS SIGN AT BASEL9E4 ,NO
BASEL9E 9DICATOR IS NE$$ FOLL[+ ! 8;B
SQU>$0 & 8;C SQU>$0 2C EA* IS FOLL[$ BY
A SPACE4

    ,EXP&$ ,USE ( ,SUP]SCRIPTS
  ,9 HI< S*OOL1 /UD5TS 2G9 US+
SUP]SCRIPTS & SUBSCRIPTS 9 M WAYS ?AN
EXPLA9$ 9 PREVI\S *APT]S "<,COMMON ,CORE
,/ATE ,/&>DS ,9ITIATIVE1 ,APP5DIX ,A1
#BJAJ">4
  ,! SUP]SCRIPT 9DICATOR IS AL NE$$ TO
WRITE ! NOTA;N US$ = ! 9V]SE ( A FUNC;N4
,9 BRL1 ? IS WRITT5 ;F SUP]SCRIPT
NEGATIVE #A BASEL9E 9DICATOR & !N AN ;X
9 P>5!SES4 ,9 ? EXAMPLE1 ! NEGATIVE #A
IS N 3SID]$ TO 2 AN EXPON5T1 B 9/1D1
REF]S TO ! 9V]SE ( ! FUNC;N4         #DB
               ,*APT] #H          A#H4BD

_% F^-1"(X) _:
  ,R1D3 ;F 9V]SE ( ;X

  ,IF AN EXPON5T IS RAIS$ TO ANO!R
EXPON5T1 ! SUP]SCRIPT 9DICATOR IS US$
TWICE 9/1D ( ONCE 2F ! SECOND EXPON5T
"<,RULE ,,XIII ^S#GD;B">4
  ,! BASEL9E 9DICATOR IS N REQUIR$ 2C !
D\BLE SUP]SCRIPT 9DICATOR T]M9ATES !
PREVI\S S+LE SUP]SCRIPT 9DICATOR4 ,9 O!R
^WS1 A LEVEL 9DICATOR T]M9ATES ! E6ECT (
A PREVI\S LEVEL 9DICATOR & 9ITIATES !
LEVEL IMPLI$ BY ! NEW 9DICATOR "<,RULE
,,XIII ^S#GI">4

_% E^X^^3"*3X^2 _:
  ,R1D3 ;E TO ! ;X CUB$ P[] "TS ?REE ;X
    SQU>$

    ,EXP&$ ,USE ( ,SUBSCRIPTS
  ,! ^1SUBSCRIPT ^19DICATOR "<DOTS
#E-#F"> IS US$ TO REPRES5T A L[]$ LR OR
NUMB] 9 ALL O!R CASES4 ,"! IS NO     #DC
               ,*APT] #H          B#H4BD
PR9T EQUIVAL5T = ! SUBSCRIPT 9DICATOR4
  .=;
-----------------------------------#H4BE
  ,Z DESCRIB$ 9 ,*APT] #F1 :5 SUBSCRIPTS
>E NUMB]S ASSOCIAT$ ) _! RESPECTIVE LR
OR V>IABLE1 NO SUBSCRIPT 9DICATOR OR
CORRESPOND+ BASEL9E 9DICATOR IS NE$$4 ,!
NUM]IC SUBSCRIPT SIMPLY FOLL[S ! V>IABLE
) NO SPACE OR ANY SPECIAL SYMBOL1 & !
BASEL9E IS MA9TA9$4

_% Y-Y1 .K M(X-X1) _:
  ,R1D3 ;Y M9US ;Y SUB "O EQUALS ;M OP5
    P>5!SIS ;X M9US ;X SUB #A4

  ,_M *EMI/RY =MULAS 3TA9 SUBSCRIPTS4 ,!
NUMB]S APPE>+ Z SUBSCRIPTS 9 ! *EMICAL
=MULA 9DICATE ! NUMB] ( ATOMS ( ! ELE;T
IMMLY 2F ! SUBSCRIPT4 ,IF NO SUBSCRIPT
APPE>S1 "O ATOM ( T ELE;T IS PRES5T4
,ADDI;NAL EXAMPLES = *EMI/RY & PHYSICS C
2 F.D 9 ! APP5DICES4

                                     #DD
               ,*APT] #H          A#H4BE

_% ,C6,H12,O6 _:
  ,R1D3 ;,C SUB SIX ;,H SUB TWELVE ,O
    SUB #F OR 8FRUCTOSE0

  ,SIMIL>LY1 :5 A NUM]IC SUBSCRIPT
FOLL[S A FUNC;N S* Z LOG1 NO SUBSCRIPT
9DICATOR IS US$ & A SPACE IS PLAC$ AF !
SUBSCRIPT "<,RULE ,,XIII ^S#GG">4 ,"! W
2 M 9=MA;N ON ^! TYPES ( FUNC;NS LAT] 9
! *APT]4

_% LOG3 X _:
  ,R1D3 LOG BASE ?REE ( ;X

  ,! NEXT EXAMPLE REQUIRES ! USE (
SUBSCRIPT 9DICATORS4

_% A;N .K A;N-1"+5 _:
  ,R1D3 A SUB ;N EQUALS A SUB ;N M9US "O
    BASEL9E PLUS FIVE4

  ,EXAMPLE #H4G 3TA9S F\R EXAMPLES US+
SUP]SCRIPTS & SUBSCRIPTS4            #DE
               ,*APT] #H           #H4BF
  ,9 ! F/ EXAMPLE1 NOTICE ! SUBSCRIPT
9DICATOR IS NE$$ = ! F/ SUBSCRIPT 2C !
SUBSCRIPT IS A LR4 ,AL1 ! BASEL9E IS
NE$$ AF ! EXPON5T 2C "! IS A P>5!SIS
9/1D ( A SPACE AF ! EXPON5T4
  ,9 ! SECOND EXAMPLE1 NOTICE !
SUBSCRIPT 9DICATOR IS N NE$$ 2C ! NUM]IC
SUBSCRIPT IS ASSOCIAT$ ) A LR4
  ,9 ! ?IRD EXAMPLE1 NOTICE ! SUBSCRIPT
9DICATOR IS NE$$ 2C ! SUBSCRIPT IS AF A
P>5!SIS 9/1D ( LR4
  ,9 ! F\R? EXAMPLE1 NOTICE ! SUBSCRIPT
9DICATOR IS N NE$$ 2C ! NUM]IC SUBSCRIPT
IS ASSOCIAT$ ) A FUNC;N4 ,AL1 "! IS A
SPACE AF ! SUBSCRIPT 2C X IS "P ( A
FUNC;N4

      ,EXAMPLE #H4G
_% ,S;N .K ?A1(1-R^N")/1-R# _:
  ,R1D3 ;,S SUB ;N EQUALS OP5 FRAC;N A
    SUB "O OP5 P>5!SIS "O M9US ;R TO !
    N? P[] CLOSE P>5!SIS OV] "O M9US ;R
    CLOSE FRAC;N4
_% ?D1/V1#+?D2/V2# .K T _:           #DF
               ,*APT] #H          A#H4BF
  ,R1D3 ,OP5 FRAC;N ;D SUB "O OV] ;V SUB
    "O CLOSE FRAC;N PLUS OP5 FRAC;N ;D
    SUB TWO OV] ;V SUB TWO CLOSE FRAC;N
    EQUALS ;T4
_% ,AL(,O,H);3 _:
  ,R1D3 ;,AL OP5 P>5!SIS ,,OH CLOSE
    P>5!SIS SUB ?REE OR 8ALUM9UM
    HYDROXIDE0
_% LOG2 #8 .K #3 _:
  ,R1D3 ,! LOG BASE TWO ( EI<T EQUALS
    ?REE4
-----------------------------------#H4BG

    ,EXP&$ ,USE ( ,RADICALS
  ,/UD5TS >E 9TRODUC$ TO RADICALS ) AN
9DEX 9 EI<? GRADE "<,COMMON ,CORE ,/ATE
,/&>DS ,9ITIATIVE1 #BJAJ2 ,M>YL& ,DE"P;T
( ,$UCA;N1 #BJAE">4
  ,2F GETT+ 9TO ! NOTA;N = RADICALS 9
,NEME? ,CODE1 LET'S REVIEW "S T]M9OLOGY4
,! ^1RADICAL ^1EXPRES.N R1D Z 8! N? ROOT
( ;X0 "<_% <N>X] _:"> HAS ?REE MAJOR
COMPON5TS3
                                     #DG
               ,*APT] #H          A#H4BG

#A4 ! ^1RADICAL ^1SYMBOL "<X LOOKS L A
  *ECK M>K 9 PR9T"> _% >] _:
#B4 ! ^19DEX "<! SMALL ;N TUCK$ \TSIDE !
  RADICAL SYMBOL 9 PR9T">
#C4 ! ^1RADIC& "<;X1 ! QUANT;Y WRITT5 2N
  ! HORIZONTAL B> ( ! RADICAL SYMBOL 9
  PR9T">4

  ,AN 9DEX ( #E M1NS ! FIF? ROOT4 ,AN
9DEX ( #C M1NS ! CUBE ROOT4 ,Z M5;N$ 9
,*APT] #F1 AN 9DEX ( #B IS ! SQU>E ROOT1
& ! #B IS USUALLY N WRITT5 9 PR9T OR
BRL4
  ,:5 WRIT+ A RADICAL ) AN 9DEX 9 ,NEME?
,CODE1 FOLL[ ^! FIVE SIMPLE /EPS4

#A4 ,! 9DEX-(-RADICAL 9DICATOR "<DOTS
  #A-#B-#F"> .=<
#B4 ,! 9DEX ( ! RADICAL
#C4 ,! RADICAL SYMBOL "<DOTS #C-#D-#E">
  .=>
#D4 ,! RADIC&1 VALUE 9SIDE_/"U A RADICAL
  SYMBOL1 : Y WANT TO F9D ! ROOT (   #DH
               ,*APT] #H          B#H4BG
#E4 ,! T]M9A;N 9DICATOR "<DOTS
  #A-#B-#D-#E-#F"> .=]

  ,NO NUM]IC 9DICATOR IS US$ ON ! 9DEX
OR ! RADIC& EV5 IF X IS NUM]IC4
  ,! FOLL[+ /EPS \TL9E H[ TO WRITE !
CUBE ROOT ( #BG 9 ,NEME? ,CODE3

#A4 ,9DEX-(-RADICAL 9DICATOR "<DOTS
  #A-#B-#F"> .=<
#B4 ,?REE "<DOTS #B-#E"> .=3
#C4 ,RADICAL SYMBOL "<DOTS #C-#D-#E">
  .=>
#D4 ,TW5TY-SEV5 "<DOTS #B-#C1 DOTS
  #B-#C-#E-#F"> .=27
#E4 ,T]M9A;N 9DICATOR "<DOTS
  #A-#B-#D-#E-#F"> ]

-----------------------------------#H4BH
  _% <3>27] _:
  ,! NEXT EXAMPLE HAS AN 9DEX$ RADICAL &
AN EXPON5T4

                                     #DI
               ,*APT] #H          A#H4BH

_% <6>X^2"]-7 _:
  ,R1D3 ,! SIX? ROOT ( ;X SQU>$ 5D ROOT
    M9US SEV5

  ,NOTICE ! ^WS 85D ROOT0 >E US$ S T !
R1D] "KS ! RADICAL 5DS 2F ! M9US SIGN4
,AL1 NOTICE ! BASEL9E 9DICATOR IS US$ AF
! EXPON5T 2C ! EXPON5T 0 FOLL[$ BY !
T]M9A;N 9DICATOR 9/1D ( A SPACE4
  ,EXAMPLE #H4H HAS F\R EXAMPLES (
RADICALS ) AN 9DEX4 ,NOTICE T 9 ! LA/
EXAMPLE AN OP5+ ,NEME? ,CODE 9DICATOR &
,NEME? ,CODE T]M9ATOR >E US$ 9 ! ^W
PROBLEM4

      ,EXAMPLE #H4H
_% <5>?1/32#] _:
  ,R1D3 ,! FIF? ROOT ( OP5 FRAC;N "O OV]
    ?IRTY-TWO CLOSE FRAC;N
_% <4>Y-3] _:
  ,R1D3 ,! F\R? ROOT ( ;Y M9US ?REE 5D
    ROOT
_% <6>(X-2)^6"] _:                   #EJ
               ,*APT] #H          B#H4BH
  ,R1D3 ,! SIX? ROOT ( OP5 P>5!SIS ;X
    M9US TWO CLOSE P>5!SIS TO ! SIX? P[]
    5D ROOT
-----------------------------------#H4BI
,:AT IS ! _% <3>0.125] _:8
  ,R1D3 ,:AT IS ! CUBE ROOT ( "O HUNDR$
    TW5TY-FIVE ?\S&?S8

      ,PRACTICE #H4D
    ,9T]L9E ! FOLL[+ PROBLEMS T 3TA9
  SUP]SCRIPTS1 SUBSCRIPTS1 & RADICALS4
_%
#1_4 #5^2"-<3>?1/8#]
#2_4 LN E^Y^^3 .K Y^3
#3_4 F^-1"(X) .K <4>X-5]
#4_4 (4-X^3")^2 .K #16-8X^3"+X^6
#5_4 A;N .K #2A;N-1"+7
_:





                                     #EA
               ,*APT] #H          A#H4BI

    ,USE YR BRLWRIT] TO WRITE ! FOLL[+
  PROBLEMS T 3TA9 SUP]SCRIPTS1
  SUBSCRIPTS1 & RADICALS4
      @.<,MA? 3T5T HAS BE5 SPELL$ \ TO
    ALL[ = PRACTICE4 ,SUP]- &
    SUB-SCRIPTS >E NOT$ ) 8SUP]0 &
    8SUB01 & ! 3T5T A6ECT$ BY ! LEVEL
    *ANGE IS 3NECT$ ) HYPH5S4@.>
#A4 ;F SUP]-NEGATIVE-"O OP5-P>5!SIS ;X
  CLOSE-P>5!SIS EQUALS ! F\R? ROOT ( ;X
  @.<5D ROOT@.> PLUS SEV5
#B4 LOG T5 SUP]-;X SUP]-SUP]-?REE EQUALS
  ;X SUP]-?REE
#C4 A SUB-;C EQUALS ;V SUP]-TWO OV] ;R
#D4 ,! F\R? ROOT ( EI<TY-"O ;X SUP]-EI<T
  @.<5D ROOT@.> EQUALS ?REE ;X SUP]-TWO
#E4 ;,H SUB-TWO ;,S ,O SUB-#D @.<,ALL
  ITEMS >E UNSPAC$ 9 PR9T4@.>

    ,FUNC;NS
  ,/UD5TS >E 9TRODUC$ TO DI6]5T TYPE (
FUNC;NS 9 ,ALGEBRA #A & ,ALGEBRA #B
"<,COMMON ,CORE ,/ATE ,/&>DS         #EB
               ,*APT] #H          B#H4BI
,9ITIATIVE1 ,APP5DIX ,A1 #BJAJ">4
  ,COMMON FUNC;NS 9CLUDE3

_4 LOG "<A2REVIA;N = ! COMMON LOG>I?M OR
  LOG BASE #AJ">
_4 LN "<A2REVIA;N = ! NATURAL LOG>I?M OR
  LOG BASE ;E">
_4 COS "<A2REVIA;N = COS9E">
_4 S9 "<A2REVIA;N = S9E">
_4 TAN "<A2REVIA;N = TANG5T">

-----------------------------------#H4CJ
  ,KEEP ^! PO9TS 9 M9D :5 US+ FUNC;NS3

_4 ,! A2REVIA;NS >E US$ 9 BRL J L !Y >E
  9 PR9T "<,RULE ,,XVII">4
_4 ,NO 3TRAC;NS >E US$ 9 ! FUNC;N "NS
  "<,RULE ,,IX ^S#EE;B">4
_4 ,:E!R OR N "! IS A SPACE AF ! FUNC;N
  "N 9 PR9T1 "! IS ALW A SPACE AF A
  FUNC;N "N 9 BRL "<,RULE ,,XVII
  ^S#AAI;A">4
_4 ,IF "! IS A S+LE LR 9SIDE ( A FUNC;N1
  ! ,5GLI% LR 9DICATOR IS N US$ 9    #EC
               ,*APT] #H          A#H4CJ
  ,NEME? ,CODE "<,RULE ,,IV ^S#BG;A">4
_4 ,IF A FUNC;N HAS A NUM]IC SUBSCRIPT1
  ! SUBSCRIPT 9DICATOR IS N US$ "<,RULE
  ,,XIII ^S#GG">4

  ,EXAMPLE #H4I HAS FIVE EXAMPLES (
FUNC;NS4
  ,9 ! F/ ?REE EXAMPLES1 "! IS N A SPACE
9 PR9T AF S91 COS1 OR TAN1 B "! IS A
SPACE 9 BRL4
  ,9 ! F\R? EXAMPLE1 NO SUBSCRIPT
9DICATOR IS US$ 2C ! SUBSCRIPT ON !
FUNC;N IS NUM]ICAL1 & ! SPACE IS AF !
NUM]ICAL SUBSCRIPT4
  ,9 ! FIF? EXAMPLE1 A SUBSCRIPT
9DICATOR IS US$ 2C ! SUBSCRIPT ON !
FUNC;N IS N NUM]ICAL4 ,9 ! LA/ EXAMPLE1
LN IS US$ AL;G ) ! LR ;E4 ,^! >E
COMMONLY US$ TGR4
  ,9 ! EXAMPLES1 "! IS NO SPACE :5 "! IS
A FUNC;N "N FOLL[+ A LR1 NUMB]1 FRAC;N
9DICATORS1 OR SIGN ( OP]A;N "<,RULE
,,XVII ^S#AAI;C">4
                                     #ED
               ,*APT] #H          B#H4CJ

      ,EXAMPLE #H4I
_% COS #135^.* _:
  ,R1D3 ! COS9E ( "O HUNDR$ ?IRTY-FIVE
    DEGREES
_% TAN (-,B) .K -TAN ,B _:
  ,R1D3 ,! TANG5T ( OP5 P>5!SIS NEGATIVE
    ;,B CLOSE P>5!SIS EQUALS ! NEGATIVE
    TANG5T ( ;,B4
-----------------------------------#H4CA
_% SIN #2,A .K #2SIN ,ACOS ,A _:
  ,R1D3 ,! S9E ( TWO ,A EQUALS TWO S9E (
    ,A COS9E ( ,A4
_% LOG3 #81 .K #4 _:
  ,R1D3 ,! LOG BASE ?REE ( EI<TY-"O
    EQUALS F\R4
_% LOG;B A .K ?LN A/LN B# _:
  ,R1D3 ,! LOG BASE ;B ( A EQUALS OP5
    FRAC;N ! NATURAL LOG ( A OV] !
    NATURAL LOG ( ;B CLOSE FRAC;N4

    ,GREEK ,LRS
  ,9 ,*APT] #G Y LE>N$ ! ,GREEK LR ^1PI1
: IS WRITT5 ) ! ,GREEK LR 9DICATOR   #EE
               ,*APT] #H          A#H4CA
"<DOTS #D-#F">1 FOLL[$ BY ! LR 8;P0
"<DOTS #A-#B-#C-#D">4 ,PI IS A L[]CASE
,GREEK LR (T5 US$ :5 "W+ ) CIRCLES &
TRIGONOMETRY4
  .=.P
  ,/UD5TS >E 9TRODUC$ TO O!R USES =
,GREEK LRS 9 ,ALGEBRA #A1 GEOMETRY1
,ALGEBRA #B1 & 2Y "<,COMMON ,CORE ,/ATE
,/&>DS ,9ITIATIVE1 #BJAJ2 ,M>YL& ,DE"P;T
( ,$UCA;N1 #BJAE">4 ,BO? UPP]CASE &
L[]CASE ,GREEK LRS >E (T5 US$ 9 MA?4
  ,:5 A ,GREEK LR IS CAPITALIZ$1 ! ORD]
= WRIT+ ! ,GREEK LR IS ,GREEK LR
9DICATOR "<DOTS #D-#F">1 CAPITAL
9DICATOR "<DOT #F">1 LR4 ,COMMON ,GREEK
LRS 9CLUDE3

,"NS ( ANGLES
  ALPHA "<L[]CASE">
    _% .A _:
-----------------------------------#H4CB
  2TA "<L[]CASE">
    _% .B _:
  !TA "<L[]CASE">                    #EF
               ,*APT] #H          A#H4CB
    _% .? _:
,TO REPRES5T 8*ANGE (0
  DELTA "<UPP]CASE">
    _% .,D _:
,TO REPRES5T SUMMA;N
  SIGMA "<UPP]CASE">
    _% .,S _:

  ,GREEK LRS >E (T5 US$ ) TRIG FUNC;NS
S* Z S91 COS1 & TAN4
  ,A COMPLETE LI/ ( ,GREEK SYMBOLS C 2
F.D 9 ! APP5DICES4
  ,EXAMPLE #H4AJ 3TA9S FIVE EXAMPLES )
,GREEK LRS4

      ,EXAMPLE #H4AJ
_% TAN .? .K ?SIN .?/COS .?# _:
  ,R1D3 ,! TANG5T ( !TA EQUALS OP5
    FRAC;N S9E ( !TA OV] ! COS9E ( !TA
    CLOSE FRAC;N4
,! SYMBOL _% .,S _: REPRES5TS
  SUMMA;N4
_% SIN^2 .B+COS^2 .B .K #1 _:
  ,R1D3 ,S9E SQU>$ 2TA PLUS COS9E    #EG
               ,*APT] #H          B#H4CB
    SQU>$ 2TA EQUALS "O4
-----------------------------------#H4CC
_% SIN (?.P/2#-.A) .K COS .A _:
  ,R1D3 ,! S9E ( OP5 P>5!SIS OP5 FRAC;N
    PI OV] TWO CLOSE FRAC;N M9US ALPHA
    CLOSE P>5!SIS EQUALS COS9E ( ALPHA4
_% ,E .K -?.,D,V/.,DS# _:
  ,R1D3 ;,E EQUALS NEGATIVE OP5 FRAC;N
    DELTA ;,V OV] DELTA ;S CLOSE FRAC;N4

    ,REVIEW ( ! ,FIVE-,/EP ,RULE F
    ,*APT] #G
  ,:5 TRANSCRIB+ MO/ MODIFI$ EXPRES.NS 9
,NEME? ,CODE1 ! ^1FIVE-/EP ^1RULE IS US$
"<,RULE ,,XIV1 ^S#HF">4 ,X IS CALL$ !
FIVE-/EP RULE 2C "! >E FIVE /EPS T M/ 2
US$ 9 ! SAME ORD] EA* "T AN EXPRES.N IS
MODIFI$ US+ SPECIFI$ SYMBOLS "<,RULE
,,XIV1 ^S#HF">4 ,"! IS NO PR9T EQUIVAL5T
= ! F\R MODIFICA;N 9DICATORS
"<MULTIPURPOSE 9DICATOR1 DIRECTLY OV]
9DICATOR1 DIRECTLY "U 9DICATOR1 &
T]M9A;N 9DICATOR"> US$ ) ! FIVE-/EP
RULE4                                #EH
               ,*APT] #H          A#H4CC

      ,FIVE-,/EP ,RULE
^7,/EP #A3 ,MULTIPURPOSE 9DICATOR^'
    "<DOT #E">
  .="
^1,/EP ^1#B3 ,EXPRES.N 2+ MODIFI$
^7,/EP #C3 ,DIRECTLY OV] 9DICATOR^'
    "<DOTS #A-#B-#F">
  .=<
  OR ^7,DIRECTLY "U 9DICATOR^' "<DOTS
    #A-#D-#F">
  .=%
^1,/EP ^1#D3 ,MODIFI]
^7,/EP #E3 ,T]M9A;N 9DICATOR^' "<DOTS
    #A-#B-#D-#E-#F">
  .=]
-----------------------------------#H4CD

    ,SIGMA ,NOTA;N OR ,SUMMA;N ,NOTA;N
  ,! ,GREEK LR SIGMA IS US$ 9 SIGMA
NOTA;N4 ,! SYMBOL _% .,S _: REPRES5TS
SUMMA;N & IS US$ TO REPRES5T ! SUM OR
TOTAL4 ,SIGMA NOTA;N 9CLUDES SYMBOLS BO?
ABV & 2L SIGMA S ! 8,FIVE-,/EP       #EI
               ,*APT] #H          A#H4CD
,RULE0 NE$S TO 2 US$4 ,/EPS #C & #D W 2
REP1T$1 ONCE = :AT IS 2L SIGMA & !N AG =
:AT IS ABV SIGMA4
  ,:5 BRAILL+ AN EXPRES.N 9 SIGMA
NOTA;N1 "H >E ! /EPS Y W USE4 ,NOTICE !
REPETI;N ( /EPS #C & #D4

^7,/EP #A3 ,MULTIPURPOSE 9DICATOR^'
    "<DOT #E">
  .="
^1,/EP ^1#B3 ,EXPRES.N 2+ MODIFI$
  SIGMA "<UPP]CASE"> "<DOTS #D-#F1 DOT
#F1 DOTS #B-#C-#D">
  _% .,S _:
^7,/EP #C3 ,DIRECTLY "U 9DICATOR^'
    "<DOTS #A-#D-#F">
  .=%
^1,/EP ^1#D3 ,MODIFI] "<:AT IS "U
    SIGMA">
^7,/EP #C3 ,DIRECTLY OV] 9DICATOR^'
    "<DOTS #A-#B-#F">
  .=<
^1,/EP ^1#D3 ,MODIFI] "<:AT IS OV]
    SIGMA">                          #FJ
               ,*APT] #H          B#H4CD
^7,/EP #E3 ,T]M9A;N 9DICATOR^' "<DOTS
    #A-#B-#D-#E-#F">
  .=]
_% ".,S%K .K #3<7]K^2 _:
  ,R1D3 ,! SUM F K"7#C TO #G ( ;K SQU>$

  ,:5 COMPUT+ ? EXPRES.N1 ! /UD5T WD
SQU>E EA* ( ! NUMB]S F #C TO #G1 & !N
F9D ! SUM OR TOTAL ( ^? ANSW]S4
  ,9 O!R ^WS1 ! EXPRES.N WRITT5 \ 9
EXP&$ =M WD BE3 _%

#1_4 ".,S%K .K #3<7]K^2
  .K #3^2"+4^2"+5^2"+6^2"+7^2 _:
-----------------------------------#H4CE

  ,EXAMPLE #H4AA 3TA9S TWO M EXAMPLES (
SIGMA NOTA;N4

      ,EXAMPLE #H4AA
_% ".,S%J .K #0<,=](?1/4#)^J _:
  ,R1D3 ,! SUM F ;J EQUALS #J TO 9F9;Y (
    "O-F\R? TO ! ;J P[]
_% ".,S%I .K #2<10]5I-3 _:           #FA
               ,*APT] #H          A#H4CE
  ,R1D3 ,! SUM F I EQUALS #B TO #AJ (
    #E;I M9US #C

      ,PRACTICE #H4E
    ,9T]L9E ! FOLL[+ PROBLEMS T 3TA9
  FUNC;NS1 ,GREEK LRS1 & SIGMA NOTA;N4
_%
#1_4 LOG;B XY .K LOG;B X+LOG;B Y
#2_4 ".,S%N .K #5<8]4N+2
#3_4 COS (2.B) .K #2COS^2 .B-1
#4_4 .A .K -?.P/2#
#5_4 ,I .K ?.,D,Q/.,DT# _:

    ,USE YR BRLWRIT] TO WRITE ! FOLL[+
  PROBLEMS T 3TA9 FUNC;NS1 ,GREEK LRS1 &
  SIGMA NOTA;N4
      @.<,MA? 3T5T HAS BE5 SPELL$ \ TO
    ALL[ = PRACTICE4@.>
#A4 ;,R EQUALS ;,R SUB-Z]O OP5-P>5!SIS
  "O PLUS ALPHA CAPITAL-DELTA ;,T
  CLOSE-P>5!SIS
#B4 LOG SUB-F\R SEV5 EQUALS LN SEV5 OV]
  LN #D
#C4 TAN !TA EQUALS ;Y OV] ;X         #FB
               ,*APT] #H          B#H4CE
#D4 ,CAPITAL SIGMA @.<DIRECTLY "U@.> ;K
  EQUALS Z]O @.<DIRECTLY OV]@.> 9F9;Y
  @.<5D MODIFICA;N@.> ?REE OP5-P>5!SIS
  "O OV] FIVE CLOSE-P>5!SIS SUP]-;K
#E4 2TA EQUALS TWO HUNDR$ TW5TY-FIVE
  DEGREES
-----------------------------------#H4CF

             ,*APT] ,SUMM>Y

    ,V]TICAL ,B>
_4 ,POSITIVE OR NEGATIVE NUMB]S 5CLOS$ 9
  V]TICAL B>S >E US$ TO REPRES5T
  ABSOLUTE VALUE4
_4 ,D N USE A NUM]IC 9DICATOR )9 !
  V]TICAL B>S :5 WRIT+ AN ABSOLUTE
  VALUE4
_4 ,V]TICAL B>S C 2 5L>G$ & >E US$ =
  DET]M9ANTS 9 MATRICES4
_4 ,:5 A CAPITAL LR IS PLAC$ 2T V]TICAL
  B>S X C REPRES5T ! DET]M9ANT ( A
  MATRIX4 ,AN ,5GLI% LR 9DICATOR IS N
  US$ ) A LR PLAC$ 2T V]TICAL B>S4
_4 ,A S+LE V]TICAL B> C REPRES5T     #FC
               ,*APT] #H          A#H4CF
  8S* Z0 9 SET NOTA;N4
_4 ,A V>IABLE "<LR"> PLAC$ 2T "O OR TWO
  SETS ( V]TICAL B>S C REPRES5T !
  8MAGNITUDE (0 OR ! 8NORM (40

    ,GR\P+ ,SYMBOLS
_4 ,P>5!SES >E US$ 9 FUNC;N NOTA;N4
_4 ,! FUNC;N F"<X"> IS R1D 8;F ( ;X40
_4 ,BRACKETS & P>5!SES >E US$ TO %[
  9T]VAL NOTA;N4 ,A BRACKET M1NS ! 5D
  VALUE IS 9CLUD$ & A P>5!SIS M1NS ! 5D
  VALUE IS N 9CLUD$ 9 ! 9T]VAL4
_4 ,ANGLE BRACKETS >E US$ TO REPRES5T !
  COMPON5T =M :5 "W+ ) VECTORS4
_4 ,! NUM]IC 9DICATOR IS N USUALLY US$ )
  NUMB]S 9SIDE ( ! ANGLE BRACKETS 2C !
  NUMB]S >E 3SID]$ "P ( AN 5CLOS$ LI/4
  ,AN EXCEP;N TO ? RULE IS NUMB]S 9
  ANGLE BRACKETS T H ! DEGREE SYMBOL AF
  !M4



                                     #FD
               ,*APT] #H          B#H4CF

    ,9F9;Y
_4 ,! 9F9;Y SYMBOL IS US$ TO REPRES5T T
  AN 9T]VAL 3T9UES ="E4
_4 ,! 9F9;Y SYMBOL C AL 2 NEGATIVE :
  M1NS T ! 9T]VAL 3T9UES ="E 9 !
  NEGATIVE DIREC;N4

    ,HOLL[ ,DOT
_4 ,! HOLL[ DOT IS A SUP]SCRIPT :5 US$ Z
  DEGREES1 S X IS WRITT5 ) ! SUP]SCRIPT
  9DICATOR FOLL[$ BY ! HOLL[ DOT =
  DEGREES4
_4 ,! HOLL[ DOT IS AL US$ TO REPRES5T A
  COMPOSITE FUNC;N4 ,:5 R1D+ ! HOLL[ DOT
  9 ? CASE1 SAY 8(0 :5 Y R1* ! HOLL[
-----------------------------------#H4CG
  DOT4 ,= 9/.E1 TO R1D _% F.*G _:1 SAY
  8;F ( ;G40 ,? M1NS T G"<X"> WD 2
  SUB/ITUT$ 9TO F"<X"> :5 CR1T+ !
  COMPOSITE FUNC;N4


                                     #FE
               ,*APT] #H          A#H4CG

    ,5L>G$ ,GR\P+ ,SYMBOLS
_4 ,:5 5L>G+ A GR\P+ SYMBOL1 A DOT #F IS
    US$ 2F ! LA/ CELL ( ! GR\P+ SYMBOL J
    L A DOT #F IS US$ 9 FRONT ( A LR TO
    CAPITALIZE X4
_4 ,! 5L>G$ GR\P+ SYMBOL IS US$ 9 BRL :5
    "! IS A GR\P+ SYMBOL >.D MULTIPLE
    L9ES 9 PR9T4
_4 ,>RANGE;TS ) 5L>G$ GR\P+ SYMBOLS >E
    3SID]$ SPATIAL PROBLEMS S L1VE A
    BLANK L9E ABV & 2L ! >RANGE;T4
_4 ,"! M/ 2 AT L1/ "O 5L>G$ GR\P+ SYMBOL
    ON EA* L9E ( ! >RANGE;T4
_4 ,! GR\P+ SYMBOLS M/ 2 L9$ UP
    V]TICALLY4
_4 ,"S"TS AN >RANGE;T US+ 5L>G$ GR\P+
    SYMBOLS H PROBLEM ID5TIFI]S1 "PS (
    AN EXPRES.N1 PUNCTUA;N1 ETC4 3NECT$
    TO ! >RANGE;T4 ,^! M/ 2 PLAC$ ON !
    TOP L9E ( ! >RANGE;T 9 BRL REG>D.S (
    XS LOCA;N 9 PR9T4
_4 ,FOLL[ PR9T REG>D+ ! PRES;E OR ABS;E
    ( GR\P+ SYMBOLS4                 #FF
               ,*APT] #H          B#H4CG
_4 ,5L>G$ GR\P+ SYMBOLS >E (T5 US$ =
    PIECEWISE-DEF9$ FUNC;NS1 SY/EMS (
    EQUA;NS1 MATRICES1 & DET]M9ANTS4
_4 ,= SY/EMS ( EQUA;NS1 :5 ELE;TS ( !
    SY/EM >E V]TICALLY ALIGN$ 9 PR9T1 !
    ALIGN;T IS RETA9$ 9 BRL4
_4 ,= MATRICES & DET]M9ANTS3
  _4 ,EA* 5TRY M/ 2 MOV$ Z F> LEFT Z
    POSSIBLE 9 XS COLUMN4
  _4 ,"O COLUMN ( BLANK CELLS M/ 2 LEFT
    2T ! COLUMNS ( ! >RANGE;T4
  _4 ,! NUM]IC 9DICATOR M/ 2 US$ )
    NUM]IC 5TRIES4
_4 ,D N USE 5L>G$ GR\P+ SYMBOLS 9 !
    FOLL[+ SITUA;NS3
  _4 ,TO COV] A FRAC;N
  _4 ,TO WRITE A B9OMIAL COE6ICI5T
  _4 ,TO WRITE MAT]IAL O3UPY+ A L>GE
    AM.T ( V]TICAL SPACE S* Z A SQU>E
    ROOT 9SIDE ( ANO!R SQU>E ROOT

    ,SET ,!ORY
_4 ,AN EMPTY SET1 OR NULL SET1 IS A SET
  ) NO ELE;TS 9 X4 ,"! >E TWO WAYS   #FG
               ,*APT] #H          C#H4CG
  TO REPRES5T ! EMPTY SET4 ,"O WAY IS TO
  USE A SET ( BRACES ) A SPACE 2T !M4 ,!
  O!R IS TO USE ! EMPTY SET SYMBOL4
_4 ,! UNION & 9T]SEC;N SYMBOLS >E 3SID]$
  SIGNS ( OP]A;N1 S "! >E NO SPACES ON
  EI SIDE ( ! SYMBOLS4
-----------------------------------#H4CH
_4 ,! 8IS A SUBSET (0 & 8IS AN ELE;T (0
  SYMBOLS >E 3SID]$ SIGNS ( COMP>ISON1 S
  "! %D 2 A SPACE 2F & AF ! SYMBOLS4

    ,SUP]SCRIPTS & ,SUBSCRIPTS
_4 ,! SUP]SCRIPT 9DICATOR IS US$ TO
  REPRES5T EXPON5TS1 P[]S1 OR RAIS$ "*S
  S* Z ! DEGREE SIGN OR 9V]SE4 ,X IS US$
  AF ! BASE & 2F ! ACTUAL RAIS$ "* OR
  EXPON5T4 ,"! IS N A CORRESPOND+ SIGN 9
  PR9T = ! SUP]SCRIPT 9DICATOR4
_4 ,IF A MA!MATICAL SYMBOL1 S* Z A SIGN
  ( OP]A;N1 GR\P+ SYMBOL1 T]M9A;N
  9DICATOR1 OR FRAC;N L9E1 COMES AF !
  EXPON5T1 A BASEL9E 9DICATOR IS US$ AF
  ! EXPON5T TO 9DICATE ! EXPON5T HAS
  5D$4                               #FH
               ,*APT] #H          A#H4CH
_4 ,NO BASEL9E 9DICATOR IS NE$$ IF A
  SPACE OR O!R LEVEL 9DICATOR FOLL[S A
  SUP]SCRIPT OR SUBSCRIPT4
_4 ,"! IS NO PR9T EQUIVAL5T = ! BASEL9E
  9DICATOR4
_4 ,IF AN EXPON5T IS RAIS$ TO ANO!R
  EXPON5T1 ! SUP]SCRIPT 9DICATOR IS US$
  TWICE1 9/1D ( ONCE1 2F ! SECOND
  EXPON5T4
_4 ,A LEVEL 9DICATOR T]M9ATES ! E6ECT (
  A PREVI\S LEVEL 9DICATOR & 9ITIATES !
  LEVEL IMPLI$ BY ! NEW 9DICATOR
_4 ,! SUBSCRIPT 9DICATOR IS DOTS #E-#F &
  IS US$ TO REPRES5T A L[]$ LR OR NUMB]4
  ,"! IS NO PR9T EQUIVAL5T = ! SUBSCRIPT
  9DICATOR4
_4 ,:5 SUBSCRIPTS >E NUMB]S ASSOCIAT$ )
  _! RESPECTIVE LR OR V>IABLE1 NO
  SUBSCRIPT 9DICATOR OR CORRESPOND+
  BASEL9E 9DICATOR IS NE$$4 ,! NUM]IC
  SUBSCRIPT SIMPLY FOLL[S ! V>IABLE ) NO
  SPACE OR ANY SPECIAL SYMBOL1 & !
  BASEL9E IS MA9TA9$4
_4 ,_M *EMI/RY =MULAS 3TA9           #FI
               ,*APT] #H          B#H4CH
  SUBSCRIPTS4 ,! NUMB]S APPE>+ Z
  SUBSCRIPTS 9 ! *EMICAL =MULA 9DICATE !
  NUMB] ( ATOMS ( ! ELE;T IMMLY 2F !
  SUBSCRIPT4 ,IF NO SUBSCRIPT APPE>S1 "O
  ATOM ( T ELE;T IS PRES5T4

    ,RADICALS ) & )\T AN ,9DEX
_4 ,! RADICAL SYMBOL 2G9S A SQU>E ROOT1
  & ! T]M9A;N 9DICATOR 5DS ! SQU>E ROOT4
_4 ,! NUM]IC 9DICATOR IS N US$ 2T !
  RADICAL SYMBOL & ! T]M9A;N 9DICATOR4
_4 ,"! IS NO PR9T EQUIVAL5T = ! T]M9A;N
  9DICATOR4
_4 ,! SQU>E ROOT IS ASSUM$ TO H AN 9DEX
  ( #B4
_4 ,! 9DEX-(-RADICAL-9DICATOR IS US$ =
  RADICALS ) AN 9DEX O!R ?AN #B4
-----------------------------------#H4CI
_4 ,! 9DEX IS FRAM$ 2T ! 9DEX-(-RADICAL
  9DICATOR & ! RADICAL SYMBOL4

    ,FUNC;NS
_4 ,COMMON FUNC;NS 9CLUDE LOG1 LN1 S91
  COS1 & TAN4                        #GJ
               ,*APT] #H          A#H4CI
_4 ,! #C-LR A2REVIA;NS >E US$ 9 BRL J L
  !Y >E 9 PR9T4 ,NO 3TRAC;NS >E US$ 9 !
  FUNC;N "NS4
_4 ,:E!R OR N "! IS A SPACE AF ! FUNC;N
  "N 9 PR9T1 "! IS ALW A SPACE AF A
  FUNC;N "N 9 BRL4
_4 ,IF "! IS A S+LE LR 9SIDE ( A FUNC;N1
  ! ,5GLI% LR 9DICATOR IS N US$4

    ,GREEK LRS
_4 ,! ,GREEK LR 9DICATOR IS US$ :5 WRIT+
  ,GREEK LRS4
_4 ,:5 A ,GREEK LR IS CAPITALIZ$1 ! ORD]
  = WRIT+ ! ,GREEK LR IS ,GREEK LR
  9DICATOR1 CAPITAL 9DICATOR1 LR4
_4 ,COMMON ,GREEK LRS 9CLUDE ALPHA1 2TA1
  !TA1 PI1 DELTA1 & SIGMA4
_4 ,GREEK LRS >E (T5 US$ ) TRIG FUNC;NS
  S* Z S91 COS1 & TAN4

    ,SIGMA ,NOTA;N
_4 ,! ,GREEK LR SIGMA IS US$ 9 SIGMA
  NOTA;N4
_4 ,? NOTA;N IS US$ TO REPRES5T !    #GA
               ,*APT] #H          B#H4CI
  SUM OR TOTAL4
_4 ,SIGMA NOTA;N 9CLUDES SYMBOLS BO? ABV
  & 2L SIGMA1 S ! 8,FIVE-,/EP ,RULE0 M/
  2 US$4 ,/EP #C & #D W 2 REP1T$1 ONCE =
  :AT IS 2L SIGMA & !N AG = :AT IS ABV
  SIGMA4
-----------------------------------#H4DJ

    ,*APT] #H3 ,ANSW] ,KEY
      ,ANSW] #H4A
#A4 ,OP5-P>5!SIS ;G HOLL[-DOT ;F
  CLOSE-P>5!SIS OP5-P>5!SIS SIX
  CLOSE-P>5!SIS EQUALS V]TICAL-B> SIX
  V]TICAL-B> PLUS TWO
#B4 ,OP5-BRACKET NEGATIVE EI<T1 9F9;Y
  CLOSE-P>5!SIS
#C4 ,OP5-BRACE ;Y V]TICAL-B> ;Y IS LESS
  ?AN N9E CLOSE-BRACE
#D4 ,OP5 ANGLE-BRACKET SEV51 ?IRTY
  DEGREES CLOSE ANGLE-BRACKET
#E4 ,V]TICAL-B> ;,M V]TICAL-B> EQUALS
  NEGATIVE T5

                                     #GB
               ,*APT] #H          A#H4DJ

_%
#1_4 .(X \ X .1: #0.)
#2_4 \,C\ .K AD-BC
#3_4 (H.*F.*G)(X) .K >X-2]+1
#4_4 (-2, 9@)
#5_4 ..(3, 90^.*"..) _:

      ,ANSW] #H4B
#A4 ,OP5-P>5!SIS SIX @.<DIRECTLY "U@.>
  TWO CLOSE-P>5!SIS
#B4 ;G OP5-P>5!SIS ;X CLOSE-P>5!SIS
  EQUALS 5L>G$ OP5-BRACKET @.<L9E #A@.>
  TWO ;X M9US "O1 = ;X IS LESS ?AN F\R
  @.<L9E #B@.> FIVE M9US ;X1 = ;X IS
  GRT] ?AN OR EQUAL TO F\R
#C4 ,5L>G$ OP5-BRACE @.<L9E #A@.> TWO ;X
  M9US ?REE ;Y EQUALS SIX @.<L9E #B@.>
  NEGATIVE ?REE ;X PLUS F\R ;Y EQUALS
  TWELVE
#D4 ;,C EQUALS 5L>G$ OP5-BRACKET @.<L9E
  #A@.> SEV5 NEGATIVE "O Z]O @.<L9E
  #B@.> NEGATIVE ?REE EI<T NEGATIVE N9E
  5L>G$ CLOSE-BRACKET                #GC
               ,*APT] #H          B#H4DJ
#E4 ,5L>G$ V]TICAL-B> @.<L9E #A@.>
  NEGATIVE FIVE SIX @.<L9E #B@.>
  NEGATIVE T5 TWELVE 5L>G$ V]TICAL-B>
-----------------------------------#H4DA

      ,ANSW] #H4B "<3T9U$">
_%

#1_4 Y .K .,(-0.5X+3, X "K: -#2
          .,(7+?3/4#X, X .1: -#2

#2_4 ,(-A+5B .K #10
     ,(   7B .K #21

#3_4 ,D .K @,(#0  -#2@,)
           @,(#6  #10@,)
           @,(-#8 #1 @,)

#4_4 ,\A B,\
     ,\C D,\

#5_4 (?4/X+Y#)-(?2/X-Y#)
_:
                                     #GD
               ,*APT] #H          A#H4DA

      ,ANSW] #H4C
#A4 ,OP5-BRACE ;S1 ;T CLOSE-BRACE IS A
  SUBSET ( OP5-BRACE ;S1 ;E1 ;T
  CLOSE-BRACE
#B4 ;,Y UNION EMPTY-SET EQUALS ;,Y
#C4 SIXTE5 IS AN ELE;T ( OP5-BRACE "O1
  F\R1 N9E1 SIXTE51 TW5TY-FIVE
  CLOSE-BRACE
#D4 OP5-BRACE PIG1 C[ CLOSE-BRACE
  9T]SEC;N OP5-BRACE CAT1 DOG
  CLOSE-BRACE EQUALS OP5-BRACE
  CLOSE-BRACE @.<BLANK SPACE 2T F9AL SET
  ( BRACES@.>
#E4 ,A UNION ;,B EQUALS OP5-BRACE "O1
  ?REE1 FIVE1 FIFTE5 CLOSE-BRACE
-----------------------------------#H4DB

      ,ANSW] #H4C "<3T9U$">
_%
#1_4 .(10, 15, 20.).%.(10, 20, 30.)
  .K .(10, 20.)
#2_4 ,D _"K: ,A
                                     #GE
               ,*APT] #H          A#H4DB
#3_4 .(A, B, C.).+.(X, Y, Z.)
  .K .(A, B, C, X, Y, Z.)
#4_4 A @E .(M, A, T, H.)
#5_4 ,E.%,F .K _0 _:

      ,ANSW] #H4D
#A4 ,FIVE SUP]-TWO M9US ! CUBE ROOT ( "O
  OV] EI<T
#B4 LN ;E SUP]-;Y SUP]-SUP]-?REE EQUALS
  ;Y SUP]-?REE
#C4 ;F SUP]-NEGATIVE-"O OP5-P>5!SIS ;X
  CLOSE-P>5!SIS EQUALS ! F\R? ROOT ( ;X
  M9US FIVE @.<5D ROOT@.>
#D4 ,OP5-P>5!SIS F\R M9US ;X SUP]-?REE
  CLOSE-P>5!SIS SUP]-TWO EQUALS SIXTE5
  M9US EI<T ;X SUP]-?REE PLUS ;X
  SUP]-SIX
#E4 A SUB-;N EQUALS TWO A SUB-;N-M9US-"O
  PLUS SEV5

_%
#1_4 F^-1"(X) .K <4>X]+7
#2_4 LOG #10^X^^3 .K X^3
#3_4 A;C .K ?V^2"/R#                 #GF
               ,*APT] #H          B#H4DB
#4_4 <4>81X^8"] .K #3X^2
#5_4 ,H2,S,O4 _:
-----------------------------------#H4DC

      ,ANSW] #H4E
#A4 LOG SUB-;B XY EQUALS LOG SUB-;B ;X
  PLUS LOG SUB-;B ;Y
#B4 ,CAPITAL SIGMA @.<DIRECTLY "U@.> ;N
  EQUALS FIVE @.<DIRECTLY OV]@.> EI<T
  @.<5D MODIFICA;N@.> F\R ;N PLUS TWO
#C4 COS OP5-P>5!SIS TWO 2TA
  CLOSE-P>5!SIS EQUALS TWO COS SUP]-TWO
  2TA M9US "O
#D4 ALPHA EQUALS NEGATIVE @.<FRAC;N
  />T@.> PI OV] TWO
#E4 ,I EQUALS @.<FRAC;N />T@.>
  CAPITAL-DELTA ;,Q OV] CAPITAL-DELTA ;T

_%
#1_4 ,R .K ,R0(1+.A.,D,T)
#2_4 LOG4 #7 .K ?LN #7/LN #4#
#3_4 TAN .? .K ?Y/X#
#4_4 ".,S%K .K #0<,=]3(?1/5#)^K
#5_4 .B .K #225^.* _:                #GG