document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=225353>Question 225353</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Rationalize the denominator\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "______2____<BR>\n" );
document.write( "&#8730;[3] + &#8730;[2]<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #168170 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=168170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Rationalizing denominators means to eliminate any irrational numbers (like square roots) from the denominator <i>using proper Algebra</i>! Your expression has a binomial (two term) denominator. To rationalize binomial denominators we use the following pattern: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This pattern shows how to take a two term expression, like (a+b) or (a-b), and make it turn into an expression of perfect squares, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+-+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Pairs of expressions like (a+b) and (a-b) are called conjugates. And when you multiply conjugates you get an expression containing nothing but perfect squares.<br><BR>\n" );
document.write( "Let's use this on your problem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28sqrt%283%29+%2B+sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"2%2F%28sqrt%283%29+%2B+sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Your denominator is of the form (a-b). So we will multiply the numerator and denominator by its conjugate, (a+b):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F%28sqrt%283%29+%2B+sqrt%282%29%29%29%2A%28%28sqrt%283%29-sqrt%282%29%29%2F%28sqrt%283%29-sqrt%282%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the distributive property on top and the pattern (or FOIL) on the bottom:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282sqrt%283%29+%2B+2sqrt%282%29%29%2F%28%28sqrt%283%29%29%5E2+-+%28sqrt%282%29%29%5E2%29 ALIGN=MIDDLE  ALT=\"%282sqrt%283%29+%2B+2sqrt%282%29%29%2F%28%28sqrt%283%29%29%5E2+-+%28sqrt%282%29%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282sqrt%283%29+%2B+2sqrt%282%29%29%2F%283+-+2%29 ALIGN=MIDDLE  ALT=\"%282sqrt%283%29+%2B+2sqrt%282%29%29%2F%283+-+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282sqrt%283%29+%2B+2sqrt%282%29%29%2F1 ALIGN=MIDDLE  ALT=\"%282sqrt%283%29+%2B+2sqrt%282%29%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%283%29+%2B+2sqrt%282%29 ALIGN=MIDDLE  ALT=\"2sqrt%283%29+%2B+2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );