document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=181801>Question 181801</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (x)^1/2-(x+3)^1/2=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #136392 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=136392\">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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math x^{\frac{1}{2}} - (x + 3)^{\frac{1}{2}} = 0\">\r<BR>\n" );
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document.write( "If you first add <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 3)^{\frac{1}{2}} \"> to both sides of the equation and re-write the fractional exponents as radicals:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math sqrt{x}= sqrt{x + 3}\">\r<BR>\n" );
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document.write( "You can readily see that this equation has no solution.  That's because over the most restrictive domain (the left hand side function), namely \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \{x | x\ \in\ \R,\ x \geq 0\}\">,\r<BR>\n" );
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document.write( "The values of the two sides of the equation will differ by a factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large sqrt{3}\"> at the low end of the domain and will differ by a decreasing amount as <i>x</i> gets larger, but the difference can never be exactly zero.\r<BR>\n" );
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document.write( "This is the reason that using ordinary solution methods on this equation results in an absurdity:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math sqrt{x}= sqrt{x + 3}\">\r<BR>\n" );
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document.write( "Square both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math x = x + 3\">\r<BR>\n" );
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document.write( "Add -<i>x</i> to both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math 0 = 3\">  Absurd.\r<BR>\n" );
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document.write( "Therefore, the solution set of the given equation is the empty set.\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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