document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=246032>Question 246032</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the equation is: (radical 12 + radical 8) (radical 18 + radical 27) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #179719 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=179719\">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=%28sqrt%2812%29+%2B+sqrt%288%29%29+%28sqrt%2818%29+%2B+sqrt%2827%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%2812%29+%2B+sqrt%288%29%29+%28sqrt%2818%29+%2B+sqrt%2827%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "First of all, this is not an equation. There is no equals sign. This is an expression.<br><BR>\n" );
document.write( "To simplify this expression we can...<BR>\n" );
document.write( "1) Just go ahead and multiply it out and then simplify the square roots:<BR>\n" );
document.write( "Multiply this out, using FOIL:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2812%29sqrt%2818%29+%2B+sqrt%2812%29sqrt%2827%29+%2B+sqrt%288%29sqrt%2818%29+%2B+sqrt%288%29sqrt%2827%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28216%29+%2B+sqrt%28324%29+%2B+sqrt%28144%29+%2B+sqrt%28216%29 ALIGN=MIDDLE  ALT=\"sqrt%28216%29+%2B+sqrt%28324%29+%2B+sqrt%28144%29+%2B+sqrt%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28324%29+=+18 ALIGN=MIDDLE  ALT=\"sqrt%28324%29+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28144%29+=+12 ALIGN=MIDDLE  ALT=\"sqrt%28144%29+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Simplifying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28216%29 ALIGN=MIDDLE  ALT=\"sqrt%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not as easy. 216 is not a perfect square. But it does have one or more perfect square factors. The quickest way to simplify <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28216%29 ALIGN=MIDDLE  ALT=\"sqrt%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is to find its largest perfect square factor (which turns out to be 36). But it may be hard to see this because, after all, who knows 36's multiplication facts? So I'll do this in a way you should be able to see yourself. 4 is a perfect square and it is a factor of 216. So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28216%29+=+sqrt%284%2A54%29+=+sqrt%284%29%2Asqrt%2854%29+=+2sqrt%2854%29 ALIGN=MIDDLE  ALT=\"sqrt%28216%29+=+sqrt%284%2A54%29+=+sqrt%284%29%2Asqrt%2854%29+=+2sqrt%2854%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Of course we keep simplifying square roots until cannot do it anymore. Are there any perfect square factors of 54? Yes, 9. So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28216%29+=+sqrt%284%2A54%29+=+sqrt%284%29%2Asqrt%2854%29+=+2sqrt%2854%29+=+2sqrt%289%2A6%29+=+2sqrt%289%29%2Asqrt%286%29+=+2%2A3%2Asqrt%286%29+=+6sqrt%286%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are no perfect square factors of 6 (other than 1) so there is nothing else we can do to simplify <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28216%29 ALIGN=MIDDLE  ALT=\"sqrt%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28216%29+%2B+sqrt%28324%29+%2B+sqrt%28144%29+%2B+sqrt%28216%29 ALIGN=MIDDLE  ALT=\"sqrt%28216%29+%2B+sqrt%28324%29+%2B+sqrt%28144%29+%2B+sqrt%28216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6sqrt%286%29+%2B+18+%2B+12+%2B+6sqrt%286%29 ALIGN=MIDDLE  ALT=\"6sqrt%286%29+%2B+18+%2B+12+%2B+6sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding like terms we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30+%2B+12sqrt%286%29 ALIGN=MIDDLE  ALT=\"30+%2B+12sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "2) Simplify square roots, then multiply, then simplify square roots again.<BR>\n" );
document.write( "This may seem like it would be longer. But maybe not because by simplifying square roots at the beginning we end up with smaller, easier to use numbers after we multiply. Since we've already seen examples of simplifying square roots above I will not explain it again here:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2812%29+%2B+sqrt%288%29%29+%28sqrt%2818%29+%2B+sqrt%2827%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%2812%29+%2B+sqrt%288%29%29+%28sqrt%2818%29+%2B+sqrt%2827%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The square roots simplify to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282sqrt%283%29+%2B+2sqrt%282%29%29%283sqrt%282%29+%2B+3sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%282sqrt%283%29+%2B+2sqrt%282%29%29%283sqrt%282%29+%2B+3sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%283%29%2A3sqrt%282%29+%2B+2sqrt%283%29%2A3sqrt%283%29+%2B+2sqrt%282%29%2A3sqrt%282%29+%2B+2sqrt%282%29%2A3sqrt%283%29 ALIGN=MIDDLE   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=6%2Asqrt%286%29+%2B+6%2A3+%2B+6%2A2+%2B+6%2Asqrt%286%29 ALIGN=MIDDLE  ALT=\"6%2Asqrt%286%29+%2B+6%2A3+%2B+6%2A2+%2B+6%2Asqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Asqrt%286%29+%2B+18+%2B+12+%2B+6%2Asqrt%286%29 ALIGN=MIDDLE  ALT=\"6%2Asqrt%286%29+%2B+18+%2B+12+%2B+6%2Asqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As we saw before <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29+ ALIGN=MIDDLE  ALT=\"sqrt%286%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot be simplified further. So we just add like terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30+%2B+12sqrt%286%29 ALIGN=MIDDLE  ALT=\"30+%2B+12sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Even though the second approach has three steps and the first approach has only two steps, I hope you see that the second approach can be much easier and faster because of the smaller numbers involved.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );