document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=243988>Question 243988</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove cube root 6 is irrational </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178619 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=178619\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Prove <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt[3]{6}\"> is irrational.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Assume <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt[3]{6}\"> is rational.  That means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \exists a,\,b\ \in\ Z\ |\ \frac{a}{b}\ =\ \sqrt[3]{6}\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{a}{b}\"> is reduced to lowest terms, that is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\"> have no common integer factors.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It follows then that:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{a^3}{b^3}\ =\ 6\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a^3\ =\ 6b^3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since at least one factor of the RHS is even, the entire RHS must be even.  Since the RHS is even, the LHS must therefore also be even.  Since the product of two odd numbers is always odd, it follows that:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a^3\, \text{mod}\,2 =\ 0\ \Rightarrow\ a^2\, \text{mod}\, 2\ =\ 0\ \Rightarrow\ a\, \text{mod}\, 2\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> is even, it follows that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \exists\ c\ |\ c\ =\ \frac{a}{2}\ \Rightarrow\ 2c\ =\ a\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (2c)^3\ =\ 6b^3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2^2c^3\ =\ 3b^3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, since at least one factor of the LHS is even, the LHS must be even.  Therefore the RHS must be even and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b^3\"> must be even.  But if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b^3\"> is even, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\"> must be even.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the the original assumption leads to the conclusion that both <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\"> are even, contradicting the part of the original assumption that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\"> have no common integer factors, the assumption that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt[3]{6}\"> is rational must be false.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt[3]{6}\"> is irrational.  QED.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );