document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1134898>Question 1134898</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Why is it NOT necessary for two radical expressions have the same index if they are to be multiplied? Provide your own example including correct answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #752399 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=752399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "One way to look at this is to take two radicals: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cb%29 ALIGN=MIDDLE  ALT=\"root%283%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which differing indices. \r<BR>\n" );
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document.write( "But we can introduce the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LCM+ ALIGN=MIDDLE  ALT=\"LCM+\"   DISABLED_style= \"max-width:90%; height:auto;\" >of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and write <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29=root%286%2Ca%5E3%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29=root%286%2Ca%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cb%29=root%286%2Cb%5E2%29 ALIGN=MIDDLE  ALT=\"root%283%2Cb%29=root%286%2Cb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=index ALIGN=MIDDLE  ALT=\"index\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This introduces a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=common ALIGN=MIDDLE  ALT=\"common\"   DISABLED_style= \"max-width:90%; height:auto;\" > radical  -  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >th root.\r<BR>\n" );
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document.write( " Now we multiply under the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=same ALIGN=MIDDLE  ALT=\"same\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+index ALIGN=MIDDLE  ALT=\"+index\"   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=sqrt%28a%29%2Aroot%283%2Cb%29=root%286%2Ca%5E3%29%2Aroot%286%2Cb%5E2%29=root%286%2Ca%5E3%2Ab%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So in the general case we need the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LCM ALIGN=MIDDLE  ALT=\"LCM\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the indices as the<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+common ALIGN=MIDDLE  ALT=\"+common\"   DISABLED_style= \"max-width:90%; height:auto;\" > radical and then we use the power indices under the common radical. \r<BR>\n" );
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document.write( "So it is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=not ALIGN=MIDDLE  ALT=\"not\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+necessary ALIGN=MIDDLE  ALT=\"+necessary\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > radical expressions to have the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=same ALIGN=MIDDLE  ALT=\"same\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+index ALIGN=MIDDLE  ALT=\"+index\"   DISABLED_style= \"max-width:90%; height:auto;\" > in order to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=multiply ALIGN=MIDDLE  ALT=\"multiply\"   DISABLED_style= \"max-width:90%; height:auto;\" > them.\r<BR>\n" );
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document.write( "NUMERICAL EXAMPLE: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28121%29 ALIGN=MIDDLE  ALT=\"sqrt%28121%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C27%29 ALIGN=MIDDLE  ALT=\"root%283%2C27%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=sqrt%28121%29=11 ALIGN=MIDDLE  ALT=\"sqrt%28121%29=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C27%29=3 ALIGN=MIDDLE  ALT=\"root%283%2C27%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " We know the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2A3=33 ALIGN=MIDDLE  ALT=\"11%2A3=33\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "But let's see if we get the same result by putting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=121 ALIGN=MIDDLE  ALT=\"a=121\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b=27 ALIGN=MIDDLE  ALT=\"+b=27\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the formula above.\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28121%29%2Aroot%283%2C27%29 ALIGN=MIDDLE  ALT=\"sqrt%28121%29%2Aroot%283%2C27%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=root%286%2C%2811%5E2%29%5E3%29%2Aroot%286%2C%283%5E3%29%5E2%29 ALIGN=MIDDLE  ALT=\"root%286%2C%2811%5E2%29%5E3%29%2Aroot%286%2C%283%5E3%29%5E2%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=root%286%2C177561%2A729%29%7D%7D%0D%0A%0D%0A=%7B%7B%7B11%2A3 ALIGN=MIDDLE  ALT=\"root%286%2C177561%2A729%29%7D%7D%0D%0A%0D%0A=%7B%7B%7B11%2A3\"   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=33 ALIGN=MIDDLE  ALT=\"33\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the formula works.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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