document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209872>Question 1209872</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify \frac{6 \sqrt{2]}{\sqrt{2} - \sqrt{3} - \sqrt{5}} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850946 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850946\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To simplify the given expression, we need to rationalize the denominator.\r<BR>\n" );
document.write( "\n" );
document.write( "Let the expression be denoted by $E$:<BR>\n" );
document.write( "$$E = \frac{6\sqrt{2}}{\sqrt{2} - \sqrt{3} - \sqrt{5}}$$\r<BR>\n" );
document.write( "\n" );
document.write( "First, multiply the numerator and denominator by $(\sqrt{2} - \sqrt{3} + \sqrt{5})$:<BR>\n" );
document.write( "$$E = \frac{6\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})}{(\sqrt{2} - \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} + \sqrt{5})}$$\r<BR>\n" );
document.write( "\n" );
document.write( "The denominator is of the form $(a-b)(a+b) = a^2 - b^2$, where $a = \sqrt{2} - \sqrt{3}$ and $b = \sqrt{5}$.\r<BR>\n" );
document.write( "\n" );
document.write( "So, the denominator becomes:<BR>\n" );
document.write( "$$(\sqrt{2} - \sqrt{3})^2 - (\sqrt{5})^2 = (2 - 2\sqrt{6} + 3) - 5 = 5 - 2\sqrt{6} - 5 = -2\sqrt{6}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Thus,<BR>\n" );
document.write( "$$E = \frac{6\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})}{-2\sqrt{6}} = \frac{3\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})}{-\sqrt{6}}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Multiply the numerator and denominator by $\sqrt{6}$:<BR>\n" );
document.write( "$$E = \frac{3\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})(-\sqrt{6})}{6} = -\frac{\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})\sqrt{6}}{2}$$<BR>\n" );
document.write( "$$E = -\frac{\sqrt{12} - \sqrt{18} + \sqrt{30}}{2} = -\frac{2\sqrt{3} - 3\sqrt{2} + \sqrt{30}}{2}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore,<BR>\n" );
document.write( "$$E = \frac{3\sqrt{2} - 2\sqrt{3} - \sqrt{30}}{-2} = \frac{-3\sqrt{2} + 2\sqrt{3} + \sqrt{30}}{2}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Thus,<BR>\n" );
document.write( "$$E = \frac{2\sqrt{3} - 3\sqrt{2} + \sqrt{30}}{2}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{\frac{2\sqrt{3} - 3\sqrt{2} + \sqrt{30}}{2}}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );