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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 35 
"&^ - \"elementary\" fractional powers" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 26 18 "Calling sequence: " }}{PARA 0 "" 0 "
" {MPLTEXT 0 21 7 "x &^ p;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 4 "x
, p" }{TEXT -1 51 " - any algebraic expressions.  Numerical values of \+
" }{MPLTEXT 0 21 1 "p" }{TEXT -1 28 " should be rational numbers." }}}
{SECT 0 {PARA 0 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 154 "Maple's mathematics is \+
largely based on complex numbers rather than real numbers.  It uses th
e \"principal branch\" of fractional powers. In particular, if " }
{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 31 " is a negative real number a
nd " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 29 " is real but not an in
teger, " }{XPPEDIT 18 0 "z^p" "6#)%\"zG%\"pG" }{TEXT -1 184 " is not r
eal.  That is not the branch that is generally used in more elementary
 mathematics, where, for example, the cube root of a negative number w
ould be taken to be negative.  The " }{MPLTEXT 0 21 4 "surd" }{TEXT 
-1 173 " function can be used to obtain these \"elementary\" roots, bu
t it is comparatively clumsy notation and does not cover fractional ex
ponents with numerators other than 1.  The " }{MPLTEXT 0 21 2 "&^" }
{TEXT -1 52 " operator is intended to address these shortcomings." }}
{PARA 15 "" 0 "" {XPPEDIT 18 0 "x &^ p" "6#-%#&^G6$%\"xG%\"pG" }{TEXT 
-1 23 " is always a branch of " }{XPPEDIT 18 0 "x^p" "6#)%\"xG%\"pG" }
{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 13 "For positive " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x &^ p" "
6#-%#&^G6$%\"xG%\"pG" }{TEXT -1 16 " is the 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 44 " is a fraction we have the following
 cases: " }}{PARA 15 "" 0 "" {TEXT -1 3 "-  " }{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," }}{PARA 15 "" 0 "" 
{TEXT -1 2 "- " }{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," }}{PARA 15 "" 0 "" {TEXT -1 2 "- " }{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$%\"xG*&%\"mG\"\"\"%
\"nG!\"\"" }{TEXT -1 12 " is complex." }}{PARA 15 "" 0 "" {TEXT -1 5 "
When " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 30 " is not known to be \+
real, and " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 16 " is a fraction,
 " }{XPPEDIT 18 0 "x&^p" "6#-%#&^G6$%\"xG%\"pG" }{TEXT -1 24 " is writ
ten as a surd.  " }}{PARA 15 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 
0 "p" "6#%\"pG" }{TEXT -1 57 " is given in decimal form, it is convert
ed to a fraction." }}{PARA 15 "" 0 "" {TEXT -1 89 "This operator is pa
rt of the Maple Advisor Database, and must be read in before use with \+
" }{MPLTEXT 0 21 14 "readlib(`&^`);" }{TEXT -1 24 " (note the back quo
tes)." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "readlib(`&^`);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#R6$%\"xG%\"pG6\"6$%XMaple~Advisor~Database~1.00~for~Map
le~V~Release~4~and~5G%gnCopyright~(c)~1998~by~Robert~B.~Israel.~~All~r
ights~reservedGF'@)5-%%typeG6$9%%(integerG-%#isG6#1\"\"!9$)F7F0-F.6$F0
%&floatG-%#&^G6$F7-%(convertG6$F0%)rationalG-F.FA@%-F.6$-%&numerG6#F0%
%evenG))F7FG*&\"\"\"FN-%&denomGFI!\"\"-%%surdG6$FLFO.-F=6$F7F0F'F'F'" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalc((-1)^(1/3)), surd(-
1,3), (-1)&^(1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&#\"\"\"\"\"#F%
*&%\"IGF%-%%sqrtG6#\"\"$\"\"\"F$!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "evalc((-1)^(1/4)),\nevalc(surd(-1,4)), evalc((-1)&^(1
/4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&*$-%%sqrtG6#\"\"#\"\"\"#\"
\"\"F(*&%\"IGF+F%F)F*,&F$#!\"\"F(F,F*F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "evalc((-1)^(2/3)), (-1) &^(2/3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,&#!\"\"\"\"#\"\"\"*&%\"IGF'-%%sqrtG6#\"\"$\"\"\"#F'F&F
'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalc((-1)^(5/8)),eval
c((-1)&^(5/8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*$-%%sqrtG6#,&\"
\"#\"\"\"*$-F&6#F)\"\"\"!\"\"F.#F/F)*&%\"IGF*-F&6#,&F)F*F+F*F.#F*F),&*
$F3F.F0*&F2F.F%F.F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "assu
me(n<0): n &^(3/5), n &^ (4/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*
$),$*$)%#n|irG\"\"$\"\"\"!\"\"#\"\"\"\"\"&F+F,*$)*$)F)\"\"%F+F-F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(x &^ (2/3), x = -1 .. 1
);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "plot(x &^ (3/5), x = -1 .. 1);" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "See also:" }
{TEXT -1 1 " " }{HYPERLNK 17 "^" 2 "^" "" }{TEXT -1 1 "," }{TEXT -1 1 
" " }{HYPERLNK 17 "Fractional powers of negative numbers" 2 "Fractiona
l_powers_of_negative_numbers" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "surd
" 2 "surd" "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 24 "Maple Advisor Da
tabase  " }{TEXT -1 14 "R. Israel 1998" }}}}{MARK "2 5 1 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }