document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209541>Question 1209541</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x = (16 - 8√5)^(1/3) + (16 + 8√5)^(1/3)<BR>\n" );
document.write( "A = x³ + 12x - 31<BR>\n" );
document.write( "Find A²⁰²⁵<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849389 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849389\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816-8sqrt%285%29%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"%2816-8sqrt%285%29%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816%2B8sqrt%285%29%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"%2816%2B8sqrt%285%29%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are to be simple expressions, then each of them needs to be of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"a%2Bb%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%2Asqrt%285%29%5E3%29=a%5E3%2B3a%5E2b%2Asqrt%285%29%2B3a%285b%5E2%29%2B5b%5E3sqrt%285%29=%28a%5E3%2B15ab%5E2%29%2B%283a%5E2b%2B5b%5E3%29%2Asqrt%285%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We need to have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2B15ab%5E2=16 ALIGN=MIDDLE  ALT=\"a%5E3%2B15ab%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%5E2b%2B5b%5E3=8 ALIGN=MIDDLE  ALT=\"3a%5E2b%2B5b%5E3=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "By inspection, these equations are satisfied when a=b=1. So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816-8sqrt%285%29%29%5E%281%2F3%29=1-sqrt%285%29 ALIGN=MIDDLE  ALT=\"%2816-8sqrt%285%29%29%5E%281%2F3%29=1-sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816%2B8sqrt%285%29%29%5E%281%2F3%29=1%2Bsqrt%285%29 ALIGN=MIDDLE  ALT=\"%2816%2B8sqrt%285%29%29%5E%281%2F3%29=1%2Bsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816-8sqrt%285%29%29%5E%281%2F3%29%2B%2816%2B8sqrt%285%29%29%5E%281%2F3%29=%281-sqrt%285%29%29%2B%281%2Bsqrt%285%29%29=2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=x%5E3%2B12x-31=2%5E3%2B12%282%29-31=8%2B24-31=1 ALIGN=MIDDLE  ALT=\"A=x%5E3%2B12x-31=2%5E3%2B12%282%29-31=8%2B24-31=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And finally<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2025=1%5E2025=1 ALIGN=MIDDLE  ALT=\"A%5E2025=1%5E2025=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: 1<br>\r<BR>\n" );
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