document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118001>Question 118001</A>: <I><SPAN STYLE=\"word-break: break-all;\"> express 5&#8730;72 in simplest radical form<BR>\n" );
document.write( "&#8730; = radical sign<BR>\n" );
document.write( "find the sum of: &#8730;120 + &#8730;75<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86045 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=86045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> #1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Asqrt%2872%29 ALIGN=MIDDLE  ALT=\"5%2Asqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "Let's simplify <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2872%29 ALIGN=MIDDLE  ALT=\"sqrt%2872%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%2872%29 ALIGN=MIDDLE  ALT=\"sqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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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. When you take the square root of this perfect square, you will get a rational number.\r<BR>\n" );
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document.write( "So let's list the factors of 72\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\r<BR>\n" );
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document.write( "Notice how 36 is the largest perfect square, so lets factor 72 into 36*2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2836%2A2%29 ALIGN=MIDDLE  ALT=\"sqrt%2836%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 72 into 36*2<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2836%29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%2836%29%2Asqrt%282%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%282%29 ALIGN=MIDDLE  ALT=\"6%2Asqrt%282%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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2872%29 ALIGN=MIDDLE  ALT=\"sqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now reintroduce the 5 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A6%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"5%2A6%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now multiply to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"30%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Asqrt%2872%29 ALIGN=MIDDLE  ALT=\"5%2Asqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"30%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "----------------------------<BR>\n" );
document.write( "Check:\r<BR>\n" );
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document.write( "Notice if we evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Asqrt%2872%29 ALIGN=MIDDLE  ALT=\"5%2Asqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a calculator we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Asqrt%2872%29=42.4264068711929 ALIGN=MIDDLE  ALT=\"5%2Asqrt%2872%29=42.4264068711929\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and if we evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"30%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
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document.write( "This shows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Asqrt%2872%29=30%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"5%2Asqrt%2872%29=30%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this verifies our answer \r<BR>\n" );
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document.write( "#2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28120%29%2Bsqrt%2875%29 ALIGN=MIDDLE  ALT=\"sqrt%28120%29%2Bsqrt%2875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2815%2A8%29%2Bsqrt%2815%2A5%29 ALIGN=MIDDLE  ALT=\"sqrt%2815%2A8%29%2Bsqrt%2815%2A5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 120 into 15*8 and factor 75 into 15*5\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2815%29%2Asqrt%288%29%2Bsqrt%2815%29%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%2815%29%2Asqrt%288%29%2Bsqrt%2815%29%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the square roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2815%29%28sqrt%288%29%2Bsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"sqrt%2815%29%28sqrt%288%29%2Bsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2815%29 ALIGN=MIDDLE  ALT=\"sqrt%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28120%29%2Bsqrt%2875%29 ALIGN=MIDDLE  ALT=\"sqrt%28120%29%2Bsqrt%2875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2815%29%28sqrt%288%29%2Bsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"sqrt%2815%29%28sqrt%288%29%2Bsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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