document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=74896>Question 74896</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can anyone help me with this?\r<BR>\n" );
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document.write( "My problem is how to rationalize the denominator:\r<BR>\n" );
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document.write( "2/ sqrt[6] - sqrt[5] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #53818 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=53818\">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=2%2F+%28sqrt%286%29+-+sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"2%2F+%28sqrt%286%29+-+sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "I'm going to assume that both the square root terms are in the denominator as shown above.<BR>\n" );
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document.write( "You can rationalize the denominator through using the identity:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b%29%2A%28a+-+b%29+=+a%5E2+-+b%5E2 ALIGN=MIDDLE  ALT=\"%28a+%2B+b%29%2A%28a+-+b%29+=+a%5E2+-+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can multiply out the left side of this identity just to convince yourself that the result<BR>\n" );
document.write( "equals the right side.<BR>\n" );
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document.write( "The denominator of the problem is of the form (a - b).  Suppose that we multiply the entire<BR>\n" );
document.write( "term of the problem by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b%29%2F%28a+%2B+b%29 ALIGN=MIDDLE  ALT=\"%28a+%2B+b%29%2F%28a+%2B+b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is equivalent to multiplying the term <BR>\n" );
document.write( "in the problem by 1 since the numerator of this multiplier equals the denominator.  <BR>\n" );
document.write( "I used \"a\" and \"b\" so it might be easier to see what we're trying to do.  Actually, we're <BR>\n" );
document.write( "going to let:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+sqrt%286%29 ALIGN=MIDDLE  ALT=\"a+=+sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%285%29 ALIGN=MIDDLE  ALT=\"b+=+sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "So when we multiply the original problem by  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b%29%2F%28a+%2B+b%29 ALIGN=MIDDLE  ALT=\"%28a+%2B+b%29%2F%28a+%2B+b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we're actually going<BR>\n" );
document.write( "to multiply it by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%286%29%2Bsqrt%285%29%29%2F%28sqrt%286%29%2Bsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%286%29%2Bsqrt%285%29%29%2F%28sqrt%286%29%2Bsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Let's do it:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F%28sqrt%286%29+-+sqrt%285%29%29%29%2A%28%28sqrt%286%29%2Bsqrt%285%29%29%2F%28sqrt%286%29%2Bsqrt%285%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In accordance with the identity above, after multiplying the original denominator<BR>\n" );
document.write( "of the problem by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%286%29%2Bsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%286%29%2Bsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the new, rationalized denominator becomes<BR>\n" );
document.write( "the difference between the squares of the two terms that were in the original denominator:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%286%29-sqrt%285%29%29%2A%28sqrt%286%29%2Bsqrt%285%29%29+=+%28sqrt%286%29%29%5E2+-+%28sqrt%285%29%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%286%29%29%5E2+=+6 ALIGN=MIDDLE  ALT=\"%28sqrt%286%29%29%5E2+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%285%29%29%5E2+=+5 ALIGN=MIDDLE  ALT=\"%28sqrt%285%29%29%5E2+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the new, rationalized denominator<BR>\n" );
document.write( "just equals 6 - 5 or 1.<BR>\n" );
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document.write( "Now all we have left to do is to work on the numerator.<BR>\n" );
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document.write( "The numerator of the original problem is 2, but we multiplied it by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%286%29%2Bsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%286%29%2Bsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as part of the process associated with rationalizing the denominator. When we do this we get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28sqrt%286%29%2Bsqrt%285%29%29+=+2%2Asqrt%286%29+%2B+2%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2%2A%28sqrt%286%29%2Bsqrt%285%29%29+=+2%2Asqrt%286%29+%2B+2%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That's the answer ... <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%286%29%2B2%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%286%29%2B2%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the new denominator that it is over<BR>\n" );
document.write( "is just 1.<BR>\n" );
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document.write( "Hope that after looking through this you'll be familiar with using the identity to rationalize<BR>\n" );
document.write( "denominators.  It will be useful to you when you work with complex numbers.<BR>\n" );
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