document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251889>Question 251889</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Rationalize the denominator: 2-&#8730;(7) divided by &#8730;(3) - &#8730;(2)\r<BR>\n" );
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document.write( "In the examples I have I believe I am supposed to multiply the numberator and denominator by the denominator....is this the correct way to start this equation or am I missing something. \r<BR>\n" );
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document.write( "I have read through the lessons and I am still very confused. If someone could show me the steps I would greatly appreciate it as maybe then I could wrap my brain around this.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183593 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=183593\">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( "You cannot simply multiply by the denominator because that would square the denominator.  If you square the denominator, you would still have radicals in the denominator.\r<BR>\n" );
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document.write( "Let's say that either <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\"> (or both) are the square root of an integer.  Then, in general\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(a\,+\,b\right)^2\ =\ a^2\ +\ 2ab\ +\ b^2\">\r<BR>\n" );
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document.write( "would still be irrational because while the the two 2nd power terms would be rational, the 2ab term in the middle would still be irrational -- and squaring it again would just make things worse.\r<BR>\n" );
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document.write( "However, recall 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 \ \ \ \ \ \ \ \ \ \ (a\,+\,b)(a\,-\,b)\ =\ a^2\,-\,b^2\">\r<BR>\n" );
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document.write( "So, given a binomial expression in the denominator where at least one of the factors of the binomial is an irrational square root, you need to multiply the entire fraction by 1 in the form of the CONJUGATE of the denominator divided by itself.  Form the conjugate of a binomial by simply changing the sign in the middle, that is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\,-\,b\"> is the conjugate of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\,+\,b\"> (and vice versa, of course).\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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