document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209888>Question 1209888</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Compute<BR>\n" );
document.write( "\frac{\{\sqrt{3}\} - 4 \{\sqrt{5}\}}{\{\sqrt{3}\}^2 + \{\sqrt{2}\}^2}. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850988 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=850988\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's compute the given expression step-by-step.\r<BR>\n" );
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document.write( "**Understanding Fractional Parts**\r<BR>\n" );
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document.write( "The fractional part of a number $x$, denoted by $\{x\}$, is defined as $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.\r<BR>\n" );
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document.write( "**Calculating Fractional Parts**\r<BR>\n" );
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document.write( "1.  **$\{\sqrt{3}\}$:**<BR>\n" );
document.write( "    * $\sqrt{3} \approx 1.732$<BR>\n" );
document.write( "    * $\lfloor \sqrt{3} \rfloor = 1$<BR>\n" );
document.write( "    * $\{\sqrt{3}\} = \sqrt{3} - 1$\r<BR>\n" );
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document.write( "2.  **$\{\sqrt{5}\}$:**<BR>\n" );
document.write( "    * $\sqrt{5} \approx 2.236$<BR>\n" );
document.write( "    * $\lfloor \sqrt{5} \rfloor = 2$<BR>\n" );
document.write( "    * $\{\sqrt{5}\} = \sqrt{5} - 2$\r<BR>\n" );
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document.write( "3.  **$\{\sqrt{2}\}$:**<BR>\n" );
document.write( "    * $\sqrt{2} \approx 1.414$<BR>\n" );
document.write( "    * $\lfloor \sqrt{2} \rfloor = 1$<BR>\n" );
document.write( "    * $\{\sqrt{2}\} = \sqrt{2} - 1$\r<BR>\n" );
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document.write( "**Substituting into the Expression**\r<BR>\n" );
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document.write( "Now, substitute these fractional parts into the given expression:\r<BR>\n" );
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document.write( "$$\frac{\{\sqrt{3}\} - 4 \{\sqrt{5}\}}{\{\sqrt{3}\}^2 + \{\sqrt{2}\}^2} = \frac{(\sqrt{3} - 1) - 4(\sqrt{5} - 2)}{(\sqrt{3} - 1)^2 + (\sqrt{2} - 1)^2}$$\r<BR>\n" );
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document.write( "**Simplifying the Numerator**\r<BR>\n" );
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document.write( "* $(\sqrt{3} - 1) - 4(\sqrt{5} - 2) = \sqrt{3} - 1 - 4\sqrt{5} + 8 = \sqrt{3} - 4\sqrt{5} + 7$\r<BR>\n" );
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document.write( "**Simplifying the Denominator**\r<BR>\n" );
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document.write( "* $(\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2\sqrt{3} + 1 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}$<BR>\n" );
document.write( "* $(\sqrt{2} - 1)^2 = (\sqrt{2})^2 - 2\sqrt{2} + 1 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}$<BR>\n" );
document.write( "* $(\sqrt{3} - 1)^2 + (\sqrt{2} - 1)^2 = (4 - 2\sqrt{3}) + (3 - 2\sqrt{2}) = 7 - 2\sqrt{3} - 2\sqrt{2}$\r<BR>\n" );
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document.write( "**Putting it Together**\r<BR>\n" );
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document.write( "The expression becomes:\r<BR>\n" );
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document.write( "$$\frac{\sqrt{3} - 4\sqrt{5} + 7}{7 - 2\sqrt{3} - 2\sqrt{2}}$$\r<BR>\n" );
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document.write( "**Approximating the Result**\r<BR>\n" );
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document.write( "Let's approximate the values:\r<BR>\n" );
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document.write( "* $\sqrt{3} \approx 1.732$<BR>\n" );
document.write( "* $\sqrt{5} \approx 2.236$<BR>\n" );
document.write( "* $\sqrt{2} \approx 1.414$\r<BR>\n" );
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document.write( "Numerator:<BR>\n" );
document.write( "* $1.732 - 4(2.236) + 7 = 1.732 - 8.944 + 7 = -0.212$\r<BR>\n" );
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document.write( "Denominator:<BR>\n" );
document.write( "* $7 - 2(1.732) - 2(1.414) = 7 - 3.464 - 2.828 = 0.708$\r<BR>\n" );
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document.write( "Then the fraction is:\r<BR>\n" );
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document.write( "$$\frac{-0.212}{0.708} \approx -0.299435$$\r<BR>\n" );
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document.write( "**Exact Result**\r<BR>\n" );
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document.write( "$$\frac{\sqrt{3} - 4\sqrt{5} + 7}{7 - 2\sqrt{3} - 2\sqrt{2}}$$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\frac{7 + \sqrt{3} - 4 \sqrt{5}}{7 - 2 \sqrt{2} - 2 \sqrt{3}}}$<BR>\n" );
document.write( " </SPAN>\n" );
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