document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203195>Question 203195</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help rationalizing....\r<BR>\n" );
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document.write( "9 times the cubic root 2 over/divided by the cubic root 5 minus cubic root 2. <BR>\n" );
document.write( "Please help! Soory my keyboard will not allow me to type the equation.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153300 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=153300\">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=%289%2Aroot%283%2C2%29%29%2F%28root%283%2C5%29-root%283%2C2%29%29 ALIGN=MIDDLE  ALT=\"%289%2Aroot%283%2C2%29%29%2F%28root%283%2C5%29-root%283%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=root%283%2C5%29 ALIGN=MIDDLE  ALT=\"x=root%283%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=root%283%2C2%29 ALIGN=MIDDLE  ALT=\"y=root%283%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2Ay%29%2F%28x-y%29 ALIGN=MIDDLE  ALT=\"%289%2Ay%29%2F%28x-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C5%29 ALIGN=MIDDLE  ALT=\"root%283%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with \"x\" and each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C2%29 ALIGN=MIDDLE  ALT=\"root%283%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with \"y\"\r<BR>\n" );
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document.write( "Now this is where you have to get a bit creative. It's very helpful to remember that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-y%5E3=%28x-y%29%28x%5E2%2Bxy%2By%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E3-y%5E3=%28x-y%29%28x%5E2%2Bxy%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Why is this important? Because if you cube a cube root, they will both cancel out (leaving the stuff inside as a remainder).\r<BR>\n" );
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document.write( "So if we multiply both numerator and denominator by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bxy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bxy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2Ay%28x%5E2%2Bxy%2By%5E2%29%29%2F%28%28x-y%29%28x%5E2%2Bxy%2By%5E2%29%29 ALIGN=MIDDLE  ALT=\"%289%2Ay%28x%5E2%2Bxy%2By%5E2%29%29%2F%28%28x-y%29%28x%5E2%2Bxy%2By%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2Ay%28x%5E2%2Bxy%2By%5E2%29%29%2F%28x%5E3-y%5E3%29 ALIGN=MIDDLE  ALT=\"%289%2Ay%28x%5E2%2Bxy%2By%5E2%29%29%2F%28x%5E3-y%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity given above\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2Aroot%283%2C2%29%28%28root%283%2C5%29%29%5E2%2Broot%283%2C5%29%2Aroot%283%2C2%29%2B%28root%283%2C2%29%29%5E2%29%29%2F%28%28root%283%2C5%29%29%5E3-%28root%283%2C2%29%29%5E3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=root%283%2C5%29 ALIGN=MIDDLE  ALT=\"x=root%283%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=root%283%2C2%29 ALIGN=MIDDLE  ALT=\"y=root%283%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2Aroot%283%2C2%29%28%28root%283%2C5%29%29%5E2%2Broot%283%2C5%29%2Aroot%283%2C2%29%2B%28root%283%2C2%29%29%5E2%29%29%2F%285-2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Cube the cube roots (to eliminate them)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2Aroot%283%2C2%29%28%28root%283%2C5%29%29%5E2%2Broot%283%2C5%29%2Aroot%283%2C2%29%2B%28root%283%2C2%29%29%5E2%29%29%2F%283%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C2%29%28%28root%283%2C5%29%29%5E2%2Broot%283%2C5%29%2Aroot%283%2C2%29%2B%28root%283%2C2%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C2%29%28%28root%283%2C5%29%29%5E2%2Broot%283%2C5%2A2%29%2B%28root%283%2C2%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C2%29%28%28root%283%2C5%29%29%5E2%2Broot%283%2C10%29%2B%28root%283%2C2%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C2%29%28root%283%2C5%5E2%29%2Broot%283%2C10%29%2Broot%283%2C2%5E2%29%29 ALIGN=MIDDLE  ALT=\"3%2Aroot%283%2C2%29%28root%283%2C5%5E2%29%2Broot%283%2C10%29%2Broot%283%2C2%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28root%28n%2Cx%29%29%5Em=root%28n%2Cx%5Em%29 ALIGN=MIDDLE  ALT=\"%28root%28n%2Cx%29%29%5Em=root%28n%2Cx%5Em%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C2%29%28root%283%2C25%29%2Broot%283%2C10%29%2Broot%283%2C4%29%29 ALIGN=MIDDLE  ALT=\"3%2Aroot%283%2C2%29%28root%283%2C25%29%2Broot%283%2C10%29%2Broot%283%2C4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 5 to get 25 and square 2 to get 4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C2%29%2Aroot%283%2C25%29%2B3%2Aroot%283%2C2%29root%283%2C10%29%2B3%2Aroot%283%2C2%29root%283%2C4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C2%2A25%29%2B3%2Aroot%283%2C2%2A10%29%2B3%2Aroot%283%2C2%2A4%29 ALIGN=MIDDLE  ALT=\"3%2Aroot%283%2C2%2A25%29%2B3%2Aroot%283%2C2%2A10%29%2B3%2Aroot%283%2C2%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aroot%283%2C50%29%2B3%2Aroot%283%2C20%29%2B3%2Aroot%283%2C8%29 ALIGN=MIDDLE  ALT=\"3%2Aroot%283%2C50%29%2B3%2Aroot%283%2C20%29%2B3%2Aroot%283%2C8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%2Aroot%283%2C2%29%29%2F%28root%283%2C5%29-root%283%2C2%29%29=3%2Aroot%283%2C50%29%2B3%2Aroot%283%2C20%29%2B3%2Aroot%283%2C8%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This can be verified (as I have verified it myself) with a calculator. </SPAN>\n" );
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