{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "Help Head ing" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Advice: " }{TEXT -1 37 "Fr actional powers of negative numbers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 175 "Maple's mathematics is largely based on complex numbers rather than real numbers. In particular, it uses the \"principal branch\" of fractional powers: the principal branch of " }{XPPEDIT 18 0 "z^(1/n)" "6#)%\"zG*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 22 " has argument between " }{XPPEDIT 18 0 "-Pi/n" "6#,$*&%#PiG\"\"\"%\"nG !\"\"F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi/n" "6#*&%#PiG\"\"\"%\" nG!\"\"" }{TEXT -1 22 ", and the argument of " }{XPPEDIT 18 0 "(-1)^(1 /n)" "6#),$\"\"\"!\"\"*&F%F%%\"nGF&" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "Pi/n" "6#*&%#PiG\"\"\"%\"nG!\"\"" }{TEXT -1 17 ".\nSo for example: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalc((-1)^(1/5));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#,$%#PiG#\"\"\"\"\"&F**&^#F* F*-%$sinGF&F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+V*p,4)!#5$\"+CD&y(eF&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(argument(%)/Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++?!#5" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{MPLTEXT 0 21 4 "surd" }{TEXT -1 32 " function prov ides \"elementary\" " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 19 "'th r oots, i.e. if " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 26 " is a negat ive number and " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 14 " is odd, t hen " }{MPLTEXT 0 21 9 "surd(x,n)" }{TEXT -1 24 " is the real (negativ e) " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 12 "'th root of " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "surd(-8,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"# " }}}{PARA 0 "" 0 "" {TEXT -1 149 "In order to have a somewhat nicer s yntax, as well as covering rational exponents with numerators other th an 1, I have defined an infix-form operator " }{MPLTEXT 0 21 2 "&^" } {TEXT -1 47 ". This is part of the Maple Advisor Database.\027" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "(-1)&^(2/5), (-1) &^ (3/5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 13 "For positive " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "x &^ p" "6#-%#&^G6$%\"xG%\"pG" }{TEXT -1 16 " is t he same as " }{XPPEDIT 18 0 "x^p" "6#)%\"xG%\"pG" }{TEXT -1 16 ". For negative " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 ", if " } {XPPEDIT 18 0 "p =m/n" "6#/%\"pG*&%\"mG\"\"\"%\"nG!\"\"" }{TEXT -1 43 " is a fraction we have the following cases:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assume(x<0); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 11 " both odd: " }{XPPEDIT 18 0 "`&^`(x,(m/n)) = -(-x)^(m/n);" "6#/-%#&^G6$%\"xG*&%\"mG\"\"\"%\"nG!\"\",$),$F'F,*&F) F*F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x &^ (3/5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),$%#x|irG!\"\"#\"\"$\"\"&\"\"\"F( " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 7 " \+ even, " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 6 " odd: " }{XPPEDIT 18 0 "`&^`(x,m/n) = (-x)^(m/n);" "6#/-%#&^G6$%\"xG*&%\"mG\"\"\"%\"nG! \"\"),$F'F,*&F)F*F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x \+ &^ (4/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),$%#x|irG!\"\"#\"\"%\" \"&\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "m" "6#%\"mG" } {TEXT -1 6 " odd, " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 7 " even: \+ " }{XPPEDIT 18 0 "`&^`(x,(m/n));" "6#-%#&^G6$%\"xG*&%\"mG\"\"\"%\"nG! \"\"" }{TEXT -1 12 " is complex." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x &^ (3/8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)!\" \"#\"\"(\"\")\"\"\"),$%#x|irGF%#\"\"$F(F)" }}}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 30 " is not known \+ to be real, and " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 16 " is a fra ction, " }{XPPEDIT 18 0 "x&^p" "6#-%#&^G6$%\"xG%\"pG" }{TEXT -1 87 " i s written as a surd. When p is given in decimal form, it is converted to a fraction." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "x:= 'x': \+ x &^ (3/8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%surdG6$*$)%\"xG\"\"$ \"\"\"\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x &^ 0.375; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%surdG6$*$)%\"xG\"\"$\"\"\"\"\") " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(x &^ (2/3), x = -1 .. 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(x &^ (3/5), x = -1 .. 1);" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" } {TEXT -1 1 " " }{HYPERLNK 17 "^" 2 "^" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "&^" 2 "&^" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "surd" 2 "surd" "" }} }{SECT 0 {PARA 0 "" 0 "" {TEXT 26 24 "Maple Advisor Database " } {TEXT -1 15 "R. Israel, 1998" }}}}{MARK "0 20 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }