document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=276413>Question 276413</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 32x^6y^9/z^8\r<BR>\n" );
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document.write( "all under a third root sign </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #201840 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=201840\">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=root%283%2C+32x%5E6y%5E9%2Fz%5E8%29 ALIGN=MIDDLE  ALT=\"root%283%2C+32x%5E6y%5E9%2Fz%5E8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying a cube root involves finding perfect cubes. Since there is a denominator we will also want to rationalize the denominator. I'm going to make the denominator a perfect cube at the start. This will take care of the rationalizing denomiinator before we start. Since any exponent that is divisible by 3 is a perfect cube, all I need to do is change <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E8 ALIGN=MIDDLE  ALT=\"z%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E9 ALIGN=MIDDLE  ALT=\"z%5E9\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+%2832x%5E6y%5E9%2Fz%5E8%29%28z%2Fz%29%29 ALIGN=MIDDLE  ALT=\"root%283%2C+%2832x%5E6y%5E9%2Fz%5E8%29%28z%2Fz%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+32x%5E6y%5E9z%2Fz%5E9%29 ALIGN=MIDDLE  ALT=\"root%283%2C+32x%5E6y%5E9z%2Fz%5E9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can start finding perfect cubes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+2%5E3%2A4%28x%5E2%29%5E3%28y%5E3%29%5E3z%2F%28z%5E3%29%5E3%29 ALIGN=MIDDLE  ALT=\"root%283%2C+2%5E3%2A4%28x%5E2%29%5E3%28y%5E3%29%5E3z%2F%28z%5E3%29%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the properties of radicals we can separate out all the perfect cubes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28root%283%2C+2%5E3%29%2Aroot%283%2C+4%29%2Aroot%283%2C+%28x%5E2%29%5E3%29%2Aroot%283%2C+%28y%5E3%29%5E3%29%2Aroot%283%2C+z%29%29%2Froot%283%2C+%28z%5E3%29%5E3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Replacing the cube roots of the perfect cubes we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Aroot%283%2C+4%29%2Ax%5E2%2Ay%5E3%2Aroot%283%2C+z%29%29%2Fz%5E3 ALIGN=MIDDLE  ALT=\"%282%2Aroot%283%2C+4%29%2Ax%5E2%2Ay%5E3%2Aroot%283%2C+z%29%29%2Fz%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Rearranging the factors in the numerator and combining the remaining radicals we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E2y%5E3%2Aroot%283%2C+4z%29%29%2Fz%5E3 ALIGN=MIDDLE  ALT=\"%282x%5E2y%5E3%2Aroot%283%2C+4z%29%29%2Fz%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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