document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=563151>Question 563151</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let A, B, C, be integers such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28A%29+%2B+root%283%2CB%29+%2B+root%283%2CC%29 ALIGN=MIDDLE  ALT=\"sqrt%28A%29+%2B+root%283%2CB%29+%2B+root%283%2CC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer. Prove that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28A%29 ALIGN=MIDDLE  ALT=\"sqrt%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2CB%29 ALIGN=MIDDLE  ALT=\"root%283%2CB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2CC%29 ALIGN=MIDDLE  ALT=\"root%283%2CC%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are integers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #364904 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=364904\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt{a} + \sqrt[3]{b} + \sqrt[3]{c} = k\"> where k is an integer, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt[3]{b} + \sqrt[3]{c} = k - \sqrt{a}\">. Cubing both sides,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b + c + 3\sqrt[3]{b^2c} + 3\sqrt[3]{bc^2} = k^3 - 3k^2\sqrt{a} + 3ka - a\sqrt{a}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here, we can take this equation \"modulo 1\" by eliminating all the integer expressions (b,c,k^3,3ka).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3\sqrt[3]{b^2c} + 3\sqrt[3]{bc^2} \equiv -3k^2\sqrt{a} - a\sqrt{a} (mod 1)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factor LHS\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3\sqrt[3]{bc}(\sqrt[3]{b} + \sqrt[3]{c}) \equiv -3k^2\sqrt{a} - a\sqrt{a} (mod 1)\"> Note that you can replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt[3]{b} + \sqrt[3]{c}\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k - \sqrt{a}\">. Modulo 1, this is equivalent to -sqrt(a).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3\sqrt[3]{bc}(-\sqrt{a}) \equiv -3k^2\sqrt{a} - a\sqrt{a} (mod 1)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3\sqrt[3]{bc} \equiv a - 3k^2 \equiv 0 (mod 1)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here, we show that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3\sqrt[3]{bc} \equiv 0 (mod 1)\"> i.e. it is an integer. I'll let you finish the proof that sqrt{a}, sqrt[3]{b}, and sqrt[3]{c} have to be integers. Pretty daunting problem...unfortunately we cannot assume the converse of the statement is true (i.e. if ... are integers then ... is an integer).<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );