document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118065>Question 118065</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Divide. Rationalize the denominator if necessary. Then simplify each radical expression.\r<BR>\n" );
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document.write( "1) 18+ square root of 567 OVER 9\r<BR>\n" );
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document.write( "2) -9- square root of 108 OVER 3\r<BR>\n" );
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document.write( "3) 6- square root of 20 OVER 2  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86132 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=86132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first two to show you how to do these\r<BR>\n" );
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document.write( "Start with the expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2818+%2B+sqrt%28567%29%29%2F9 ALIGN=MIDDLE  ALT=\"%2818+%2B+sqrt%28567%29%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First lets reduce <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28567%29 ALIGN=MIDDLE  ALT=\"sqrt%28567%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28567%29 ALIGN=MIDDLE  ALT=\"sqrt%28567%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression<BR>\n" );
document.write( "The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. This way the perfect square will become a rational number.\r<BR>\n" );
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document.write( "So let's list the factors of 567\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1, 3, 7, 9, 21, 27, 63, 81, 189, 567\r<BR>\n" );
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document.write( "Notice how 81 is the largest perfect square, so lets break 567 down into 81*7\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2881%2A7%29 ALIGN=MIDDLE  ALT=\"sqrt%2881%2A7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 567 into 81*7<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2881%29%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"sqrt%2881%29%2Asqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the square roots using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"9%2Asqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of the perfect square 81 to get 9 <BR>\n" );
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document.write( "So the expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28567%29 ALIGN=MIDDLE  ALT=\"sqrt%28567%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "simplifies to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"9%2Asqrt%287%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=%2818+%2B+9%2Asqrt%287%29%29%2F9 ALIGN=MIDDLE  ALT=\"%2818+%2B+9%2Asqrt%287%29%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (using the technique above)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F9+%2B+9%2Asqrt%287%29%2F9 ALIGN=MIDDLE  ALT=\"18%2F9+%2B+9%2Asqrt%287%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2B+9%2Asqrt%287%29%2F9 ALIGN=MIDDLE  ALT=\"2+%2B+9%2Asqrt%287%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F9 ALIGN=MIDDLE  ALT=\"18%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   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=2+%2B+1%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"2+%2B+1%2Asqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2F9 ALIGN=MIDDLE  ALT=\"9%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2818+%2B+sqrt%28567%29%29%2F9 ALIGN=MIDDLE  ALT=\"%2818+%2B+sqrt%28567%29%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "simplifies to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2B+sqrt%287%29 ALIGN=MIDDLE  ALT=\"2+%2B+sqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<hr>\r<BR>\n" );
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document.write( "#2\r<BR>\n" );
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document.write( "Start with the expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9+-+sqrt%28108%29%29%2F3 ALIGN=MIDDLE  ALT=\"%28-9+-+sqrt%28108%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First lets reduce <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28108%29 ALIGN=MIDDLE  ALT=\"sqrt%28108%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28108%29 ALIGN=MIDDLE  ALT=\"sqrt%28108%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression<BR>\n" );
document.write( "The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. This way the perfect square will become a rational number.\r<BR>\n" );
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document.write( "So let's list the factors of 108\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108\r<BR>\n" );
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document.write( "Notice how 36 is the largest perfect square, so lets break 108 down into 36*3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2836%2A3%29 ALIGN=MIDDLE  ALT=\"sqrt%2836%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 108 into 36*3<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2836%29%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%2836%29%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the square roots using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"6%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of the perfect square 36 to get 6 <BR>\n" );
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document.write( "So the expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28108%29 ALIGN=MIDDLE  ALT=\"sqrt%28108%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "simplifies to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"6%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9+-+6%2Asqrt%283%29%29%2F3 ALIGN=MIDDLE  ALT=\"%28-9+-+6%2Asqrt%283%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (using the technique above)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2F3+-+6%2Asqrt%283%29%2F3 ALIGN=MIDDLE  ALT=\"-9%2F3+-+6%2Asqrt%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3+-+6%2Asqrt%283%29%2F3 ALIGN=MIDDLE  ALT=\"-3+-+6%2Asqrt%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2F3 ALIGN=MIDDLE  ALT=\"-9%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   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+-+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"-3+-+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F3 ALIGN=MIDDLE  ALT=\"6%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9+-+sqrt%28108%29%29%2F3 ALIGN=MIDDLE  ALT=\"%28-9+-+sqrt%28108%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "simplifies to\r<BR>\n" );
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