document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=109192>Question 109192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use Property 1 to simplify each of the following radical expressions. Assume that all variables represent positive real numbers.\r<BR>\n" );
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document.write( "Sqrt108<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 #79629 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=79629\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=simplifying_square_roots.solver>Simplifying Square Roots (whole numbers only)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><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" );
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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.<BR>\n" );
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document.write( "  So let's list the factors of 108<BR>\n" );
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document.write( "  Factors:<BR>\n" );
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document.write( "  1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108<BR>\n" );
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document.write( "  Notice how 36 is the largest perfect square, so lets 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%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 <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;\" > simplifies to <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;\" ><BR>\n" );
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document.write( "  Check:<BR>\n" );
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document.write( "  Notice if we evaluate the square root of 108 with a calculator we get<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28108%29=10.3923048454133 ALIGN=MIDDLE  ALT=\"sqrt%28108%29=10.3923048454133\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  and if we evaluate <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;\" > we get<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Asqrt%283%29=10.3923048454133 ALIGN=MIDDLE  ALT=\"6%2Asqrt%283%29=10.3923048454133\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  This shows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28108%29=6%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%28108%29=6%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this verifies our answer  </TD></TR>\n" );
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