document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=292392>Question 292392</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Rationalize the denominator.  Assume that all variables represent positive real numbers and that the denominator is not zero.\r<BR>\n" );
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document.write( "7<BR>\n" );
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document.write( "9- (3 under a radical)\r<BR>\n" );
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document.write( "So it's 7 divided by 9 minus the square root of 3.<BR>\n" );
document.write( "I think.  I'm trying to write this so it makes sense. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #211212 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=211212\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Here's how to write it so it makes sense:\r<BR>\n" );
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document.write( "7/(9-sqrt(3))\r<BR>\n" );
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document.write( "Anyone can read that to mean\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{7}{9\ -\ \sqrt{3}}\">\r<BR>\n" );
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document.write( "First review my lesson on Rationalizing Denominators:  http://www.algebra.com/algebra/homework/Radicals/rationalizingdenominators1.lesson\r<BR>\n" );
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document.write( "Then you need the conjugate of the denominator.  If you have a binomial expression, such as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ +\ b\">, then its conjugate is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ -\ b\">\r<BR>\n" );
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document.write( "So the conjugate of your denominator is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 9\ +\ \sqrt{3}\">\r<BR>\n" );
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document.write( "Now multiply your expression by 1 in the form of the conjugate divided by itself.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\frac{7}{9\ -\ \sqrt{3}}\right)\left(\frac{9\ +\ \sqrt{3}}{9\ +\ \sqrt{3}}\right)\">\r<BR>\n" );
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document.write( "Remember that the product of a binomial and its conjugate is the difference of two squares:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{7\left(9\ +\ \sqrt{3}\right)}{81-3}\ =\ \frac{63\ +\ 7\sqrt{3}}{78}\">\r<BR>\n" );
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document.write( "Yes, it's ugly, but at least there is no radical in the denominator.\r<BR>\n" );
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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" );
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