document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=715790>Question 715790</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Rationalize the denominator \r<BR>\n" );
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document.write( "(3-Sqrt(6))/(x-(3+Sqrt(6))) \r<BR>\n" );
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document.write( "Answer should be (3x-15+6Sqrt(6)-xSqrt(6))/(x^2-6x+3) but i don't see how they are getting that answer. please explain </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #439618 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=439618\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-sqrt%286%29%29%2F%28x-%283%2Bsqrt%286%29%29%29 ALIGN=MIDDLE  ALT=\"%283-sqrt%286%29%29%2F%28x-%283%2Bsqrt%286%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First let's rewrite the denominator as additions:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-sqrt%286%29%29%2F%28x%2B%28-3%2B%28-sqrt%286%29%29%29%29 ALIGN=MIDDLE  ALT=\"%283-sqrt%286%29%29%2F%28x%2B%28-3%2B%28-sqrt%286%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The reason we changed to additions is that we can now use the Associative Property to rearrange the grouping:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-sqrt%286%29%29%2F%28%28x%2B%28-3%29%29%2B%28-sqrt%286%29%29%29%29 ALIGN=MIDDLE  ALT=\"%283-sqrt%286%29%29%2F%28%28x%2B%28-3%29%29%2B%28-sqrt%286%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A factoring 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;\" >, is often used to rationalize a multiple-term denominator. It gives us a way to to create an expression which has only perfect square terms. The reason we regrouped the denominator was so that we could more easily see the \"a\" and \"b\". If we think of the \"(x+(-3))\" as the \"a\" and the \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%286%29 ALIGN=MIDDLE  ALT=\"-sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" as the \"b\", the denominator is a+b. To use the pattern to create the perfect squares we must multiply by a-b:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283-sqrt%286%29%29%2F%28red%28%28x%2B%28-3%29%29%29%2Bblue%28%28-sqrt%286%29%29%29%29%29%28%28%28red%28x%2B%28-3%29%29%29-blue%28%28-sqrt%286%29%29%29%29%2F%28red%28%28x%2B%28-3%29%29%29-blue%28%28-sqrt%286%29%29%29%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "with the a's in red and the b's in blue. The pattern tells us that the denominator works out to be <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;\" >. In the numerator we must multiply each term of the first numerator by each term of the second numerator.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Ax%2B3%2A%28-3%29%2B3%2Asqrt%286%29%2B%28-sqrt%286%29%29%2Ax%2B%28-sqrt%286%29%29%2A%28-3%29%2B%28-sqrt%286%29%29%2A%28sqrt%286%29%29%29%2F%28red%28%28x%2B%28-3%29%29%5E2%29-blue%28%28%28-sqrt%286%29%29%5E2%29%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can use another pattern, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%29%5E2+=+a%5E2-2ab%2Bb%5E2 ALIGN=MIDDLE  ALT=\"a%2Bb%29%5E2+=+a%5E2-2ab%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B%28-3%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B%28-3%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > quickly. Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B%28-9%29%2B3sqrt%286%29%2B%28-x%2Asqrt%286%29%29%2B3sqrt%286%29%2B%28-6%29%29%2F%28%28%28x%29%5E2%2B2%28x%29%28-3%29%2B%28-3%29%5E2%29-6%29%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B%28-15%29%2B6sqrt%286%29%2B%28-x%2Asqrt%286%29%29%29%2F%28%28x%5E2%2B%28-6x%29%2B9%29-6%29%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B%28-3%29%2B6sqrt%286%29%2B%28-x%2Asqrt%286%29%29%29%2F%28x%5E2%2B%28-6x%29%2B3%29%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is the same as the answer you posted except it has additions instead of subtractions. </SPAN>\n" );
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