document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=241588>Question 241588</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify. Assume that no denominator is zero and that 0^0 is not considered. thanks\r<BR>\n" );
document.write( "\n" );
document.write( "(a7/-7b^4) ^6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #176961 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=176961\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a7%2F-7b%5E4%29%5E6 ALIGN=MIDDLE  ALT=\"%28a7%2F-7b%5E4%29%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Is that a7 a^7?<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E7%2F7b%5E4%29%5E6 ALIGN=MIDDLE  ALT=\"%28a%5E7%2F7b%5E4%29%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some reason the minus sign caused a problem<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E42%2F117649b%5E24 ALIGN=MIDDLE  ALT=\"a%5E42%2F117649b%5E24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "Note:  by convention, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%5E0+=+1 ALIGN=MIDDLE  ALT=\"0%5E0+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------------<BR>\n" );
document.write( "The following is a list of reasons why 0^0 should be 1. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Rotando & Korn show that if f and g are real functions that vanish at the origin and are analytic at 0 (infinitely differentiable is not sufficient), then f(x)^g(x) approaches 1 as x approaches 0 from the right. <BR>\n" );
document.write( "From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik): \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Some textbooks leave the quantity 0^0 undefined, because the functions 0^x and x^0 have different limiting values when x decreases to 0. But this is a mistake. We must define x^0=1 for all x , if the binomial theorem is to be valid when x=0 , y=0 , and/or x=-y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant. <BR>\n" );
document.write( "Published by Addison-Wesley, 2nd printing Dec, 1988. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As a rule of thumb, one can say that 0^0 = 1 , but 0.0^(0.0) is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0) ; but Kahan has argued that 0.0^(0.0) should be 1, because if f(x), g(x) --> 0 as x approaches some limit, and f(x) and g(x) are analytic functions, then f(x)^g(x) --> 1 . <BR>\n" );
document.write( "The discussion of 0^0 is very old. Euler argues for 0^0 = 1 since a^0 = 1 for a not equal to 0 . The controversy raged throughout the nineteenth century, but was mainly conducted in the pages of the lesser journals: Grunert's Archiv and Schlomilch's Zeitshrift. Consensus has recently been built around setting the value of 0^0 = 1 .\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );