document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=143431>Question 143431</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0130519669/algebrahomeworkh>Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Im not sure how to solve problems by rationalizing 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 #104388 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=104388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The process involves multiplying your fraction expression by 1 in a form such that when the denominators are multiplied, the denominator of the result becomes a rational number -- that is to say you have rid yourself of the radical.  The process is also known as \"Get that pesky radical OUT of my denominator\"\r<BR>\n" );
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document.write( "Example 1:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The way to make the denominator rational is to multiply it by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but we aren't allowed to change the value of the fraction.  Fortunately, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A1=a ALIGN=MIDDLE  ALT=\"a%2A1=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > no matter what <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fa=1 ALIGN=MIDDLE  ALT=\"a%2Fa=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > no matter what <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (as long as it isn't zero), so multiplying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is just another way to write 1) is allowed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fsqrt%282%29%29%28sqrt%282%29%2Fsqrt%282%29%29=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"%281%2Fsqrt%282%29%29%28sqrt%282%29%2Fsqrt%282%29%29=sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Example 2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Froot%283%2C2%29 ALIGN=MIDDLE  ALT=\"1%2Froot%283%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This one is a little tricker.  Here we need to multiply the denominator by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28root%283%2C2%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28root%283%2C2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Again, we have to multiply by 1:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Froot%283%2C2%29%29%28root%283%2C2%29%5E2%2Froot%283%2C2%29%5E2%29=%28root%283%2C2%29%5E2%29%2Froot%283%2C2%29%5E3=root%283%2C2%29%5E2%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Example 3:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsqrt%285%29%29%2F%282-sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"%281%2Bsqrt%285%29%29%2F%282-sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This one is a little trickier still.  We need to take advantage of the 'difference of two squares' factorization, that is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-b%5E2=%28a-b%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"a%5E2-b%5E2=%28a-b%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we multiply the denominator by what is called its conjugate, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2B+sqrt%282%29 ALIGN=MIDDLE  ALT=\"2+%2B+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice the sign change), then the result will be the difference of the first term squared and the second term squared with no annoying center term containing a radical.  Again, and as always, we have to multiply by 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsqrt%285%29%29%2F%282-sqrt%282%29%29%29%28%282%2Bsqrt%282%29%29%2F%282%2Bsqrt%282%29%29%29=%282%2Bsqrt%282%29%2B2sqrt%285%29%2Bsqrt%2810%29%29%2F%284-2%29=%282%2Bsqrt%282%29%2B2sqrt%285%29%2Bsqrt%2810%29%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Ewww! That's a mess, but at least the denominator is rational.  You might want to rephrase the alternate definition I gave earlier for the process to: \"Get that pesky radical OUT of my denominator and I don't care how big a mess you make in the numerator\"\r<BR>\n" );
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document.write( "Write back if you have any questions. </SPAN>\n" );
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