document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=182446>Question 182446</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify each expression. Rationalize all denominators. Assume that all variables are positive. <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 #136958 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=136958\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Rationalize the denominator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \frac {7}{1 - sqrt{3}}\">\r<BR>\n" );
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document.write( "Rationalizing a denominator means to eliminate any irrational numbers in the denominator, such as the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large sqrt{3}\"> in your example.\r<BR>\n" );
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document.write( "It won't do any good to square the denominator because you would still end up with an irrational term, so you need to remember the 'difference of two squares' factorization:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math (a + b)(a - b) = a^2 - b^2\">\r<BR>\n" );
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document.write( "So you can see that if you multiply a binomial by its conjugate (formed by using the same terms but changing the sign between them) you get the difference of the two squared terms.\r<BR>\n" );
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document.write( "In your case, the conjugate of your denominator, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1 - sqrt {3}\"> is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1 + sqrt {3}\">.\r<BR>\n" );
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document.write( "Now we can't simply multiply the denominator by its conjugate, because that would change the value of the fraction.  What we need to do is multiply the <i><b>entire fraction</b></i> by 1 in the form of:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \frac {1 + sqrt {3}}{1 + sqrt {3}} \">,\r<BR>\n" );
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document.write( "like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \left( \frac {7}{1 - sqrt{3}} \right)\left(\frac { 1 + sqrt {3} }{1 + sqrt {3}}\right) = \frac {7 (1 + sqrt {3}) }{(1 - sqrt{3})(1 + sqrt {3})} = \frac {7 + 7sqrt {3} }{1 - 3} = \frac {7 + 7sqrt {3} }{-2}\">\r<BR>\n" );
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document.write( "Now you have a nice rational number for a denominator, and therefore the expression is simpler than what you started with; the fact that the numerator is uglier than a mud fence notwithstanding.\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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