document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251882>Question 251882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem: (squareroot)x^2/18  both the numerator and denominator are under one square root sign. \r<BR>\n" );
document.write( "\n" );
document.write( "the instructions say to rationalize the denominator.\r<BR>\n" );
document.write( "\n" );
document.write( "Becasue 18's square root is 4.24 I am unsure this is the right answer (x/4.24) I believe I may be missing something. Should I factor out the denominator?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183582 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=183582\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In the first place 18's square root is NOT 4.24.  4.24 is only an approximation of the square root of 18.  You cannot exactly represent the square root of 18 with a decimal fraction.  If you could exactly represent the square root of 18 with some decimal fraction, then the square root of 18 would have to be rational -- that is because any decimal fraction with a finite number of decimal places can be represented exactly by the quotient of two integers.  But we can prove that 18 is not rational...\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first thing to do is to find the prime factorization of 18, namely 2 times 3 times 3.  Take two factors of 3 out of the radical and leave one of them on the outside, hence:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sqrt{18}\ =\ \sqrt{3\,\cdot\,3\,\cdot\,2}\ =\ \sqrt{3\,\cdot\,3}\ \cdot\ \sqrt{2}\ =\   3\sqrt{2}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now your problem looks like:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sqrt{\frac{x^2}{18}}\ =\ \frac{\sqrt{x^2}}{\sqrt{18}}\ =\ \frac{x}{\sqrt{18}}\ =\ \frac{x}{3\sqrt{2}}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So far, so good, but we still have a radical (irrational number) in the denominator.  It would be tidy if we were able to multiply the denominator by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt{2}\">, but that would change the value of the fraction.  However, we can multiply the entire fraction by 1 and not change its value.  So let's multiply by 1 in the form of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\sqrt{2}}{\sqrt{2}}\">, thus:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \left(\frac{x}{3\sqrt{2}}\right)\left(\frac{\sqrt{2}}{\sqrt{2}}\right)\ =\ \frac{x\sqrt{2}}{6}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now you have an equivalent representation of the original expression with a rational number denominator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "See my lesson on Rationalizing Denominators:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "http://www.algebra.com/algebra/homework/Radicals/rationalizingdenominators1.lesson\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );