document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=780379>Question 780379</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation:\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #475579 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=475579\">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( "No one can help you solve an equation unless you provide an equation to solve.  The word \"equation\" has a specific meaning, and that meaning is not \"a bunch of math-looking stuff\".  It means exactly this: \"Two expressions related by an equals sign.\"  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \frac{\sqrt{t}\ +\ 4}{\sqrt{t}\ -\ 4}\">\r<BR>\n" );
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document.write( "as complex as it is, is just a single expression and there is no equals sign to be seen anywhere.\r<BR>\n" );
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document.write( "Now just on the off chance that what you really meant to say is \"Please help me <i><b>simplify this expression</b></i>, then read on.\r<BR>\n" );
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document.write( "You need to rationalize the denominator.  Since you have a binomial expression in your denominator, squaring it won't do anything for you since you will still have a radical in the denominator expression.  What you need to do is multiply the denominator by the conjugate of the denominator.  What's a conjugate?  If you have a binomial <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( "The trouble is, you can't just willy-nilly multiply a denominator that is something different from 1 without changing the value of the fraction.  But you can multiply the ENTIRE fraction by 1 in the form of the conjugate of the denominator divided by itself.  Thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \left(\frac{\sqrt{t}\ +\ 4}{\sqrt{t}\ -\ 4}\right)\left(\frac{\sqrt{t}\ +\ 4}{\sqrt{t}\ +\ 4}\right)\">\r<BR>\n" );
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document.write( "This works because 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{t\ +\ 8\sqrt{t}\ +\ 16}{t\ -\ 16}\">\r<BR>\n" );
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document.write( "And since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE t\ -\ 16\"> does not evenly divide <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE t\ +\ 8\sqrt{t}\ +\ 16\">, you are done.\r<BR>\n" );
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document.write( "And, despite the fact that the finished product might <b>look</b> more complex than the original, the fact that there is no radical in the denominator makes it simpler by definition.\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" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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