document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=405428>Question 405428</A>: <I><SPAN STYLE=\"word-break: break-all;\"> cube root (8m^7n^9/n^2m^2)\r<BR>\n" );
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document.write( "i got:\r<BR>\n" );
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document.write( "cube root 8m^5n^7\r<BR>\n" );
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document.write( "8<BR>\n" );
document.write( "^<BR>\n" );
document.write( "2 *4<BR>\n" );
document.write( "     ^<BR>\n" );
document.write( "    2*2\r<BR>\n" );
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document.write( "final answer: 2cube root m^5n^7  ???? <BR>\n" );
document.write( "just want to make sure I did it right. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #286463 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=286463\">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+%288m%5E7n%5E9%29%2F%28n%5E2m%5E2%29%29 ALIGN=MIDDLE  ALT=\"root%283%2C+%288m%5E7n%5E9%29%2F%28n%5E2m%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Your first step is right on. The radicand (the expression inside the radical) simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+8m%5E5n%5E7%29 ALIGN=MIDDLE  ALT=\"root%283%2C+8m%5E5n%5E7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we look for factors of the radicand that are perfect cubes. As you already found, 8 is a perfect cube. But there are more perfect cube factors. Because of the way exponents work, the exponent on a perfect cube is <i>not a perfect cube</i> but a multiple of 3! So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E12 ALIGN=MIDDLE  ALT=\"y%5E12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E300 ALIGN=MIDDLE  ALT=\"z%5E300\"   DISABLED_style= \"max-width:90%; height:auto;\" > are all perfect cubes, even though 3, 12 and 300 are not themselves perfect cubes.<br><BR>\n" );
document.write( "So your radicand factored into as many perfect cubes as we can find is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+8m%5E3%2Am%5E2%2An%5E3%2An%5E3%2An%29 ALIGN=MIDDLE  ALT=\"root%283%2C+8m%5E3%2Am%5E2%2An%5E3%2An%5E3%2An%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For reasons that will become clear shortly I like to use the Commutative Property to rearrange the order of the factors so that all the perfect cubes are in front:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+8m%5E3%2An%5E3%2An%5E3%2Am%5E2%2An%29 ALIGN=MIDDLE  ALT=\"root%283%2C+8m%5E3%2An%5E3%2An%5E3%2Am%5E2%2An%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we use a property of radicals, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%283%2C+q%29 ALIGN=MIDDLE  ALT=\"root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%283%2C+q%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to split this cube root of a product into a product of cube roots. We want each perfect cube factor in its own cube root. The factors that are not perfect cubes can all go into one cube root:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C+8%29%2Aroot%283%2C+m%5E3%29%2Aroot%283%2C+n%5E3%29%2Aroot%283%2C+n%5E3%29%2Aroot%283%2C+m%5E2%2An%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The cube roots of the perfect cubes will simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Am%2An%2An%2Aroot%283%2C+m%5E2%2An%29 ALIGN=MIDDLE  ALT=\"2%2Am%2An%2An%2Aroot%283%2C+m%5E2%2An%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Am%2An%5E2%2Aroot%283%2C+m%5E2%2An%29 ALIGN=MIDDLE  ALT=\"2%2Am%2An%5E2%2Aroot%283%2C+m%5E2%2An%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the simplified cube root. (Note how the radical is at the end. This is the usual way to write terms like this and it is the reason I put all the perfect cubes n the front earlier.) </SPAN>\n" );
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