document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=200938>Question 200938</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is 5 different problems I've been having problems with for a while.\r<BR>\n" );
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document.write( "4/sqrt5<BR>\n" );
document.write( "4sqrtx^12/24sqrtx^12<BR>\n" );
document.write( "15sqrt8x^16/5sqrt2x^4<BR>\n" );
document.write( "(3sqrt5)(2sqrt10)<BR>\n" );
document.write( "(2sqrtx)(5sqrtx^3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #151157 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=151157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first three to get you going in the right direction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"4%2Fsqrt%285%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=%284%2Asqrt%285%29%29%2F%28sqrt%285%29%2Asqrt%285%29%29 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%285%29%29%2F%28sqrt%285%29%2Asqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both the numerator and denominator by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%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=%284%2Asqrt%285%29%29%2F%28sqrt%285%2A5%29%29 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%285%29%29%2F%28sqrt%285%2A5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Asqrt%285%29%29%2F%28sqrt%2825%29%29 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%285%29%29%2F%28sqrt%2825%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Asqrt%285%29%29%2F5 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%285%29%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 25 to get 5\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fsqrt%285%29=%284%2Asqrt%285%29%29%2F5 ALIGN=MIDDLE  ALT=\"4%2Fsqrt%285%29=%284%2Asqrt%285%29%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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=%284%2Asqrt%28x%5E12%29%29%2F%2824%2Asqrt%28x%5E12%29%29 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%28x%5E12%29%29%2F%2824%2Asqrt%28x%5E12%29%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=%284%2Asqrt%28x%5E12%29%29%2F%284%2A6%2Asqrt%28x%5E12%29%29 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%28x%5E12%29%29%2F%284%2A6%2Asqrt%28x%5E12%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 24 to get 4*6\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cross%284%29%2Asqrt%28x%5E12%29%29%2F%28cross%284%29%2A6%2Asqrt%28x%5E12%29%29 ALIGN=MIDDLE  ALT=\"%28cross%284%29%2Asqrt%28x%5E12%29%29%2F%28cross%284%29%2A6%2Asqrt%28x%5E12%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel out the common terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x%5E12%29%29%2F%286%2Asqrt%28x%5E12%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%5E12%29%29%2F%286%2Asqrt%28x%5E12%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x%5E12%29%2F%28x%5E12%29%29%2F%286%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x%5E12%29%2F%28x%5E12%29%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%29%2F6 ALIGN=MIDDLE  ALT=\"sqrt%281%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Asqrt%28x%5E12%29%29%2F%2824%2Asqrt%28x%5E12%29%29=1%2F6 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%28x%5E12%29%29%2F%2824%2Asqrt%28x%5E12%29%29=1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "# 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%29 ALIGN=MIDDLE  ALT=\"%2815%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%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=%285%2A3%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%29 ALIGN=MIDDLE  ALT=\"%285%2A3%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 15 into 5*3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cross%285%29%2A3%2Asqrt%288x%5E16%29%29%2F%28cross%285%29%2Asqrt%282x%5E4%29%29 ALIGN=MIDDLE  ALT=\"%28cross%285%29%2A3%2Asqrt%288x%5E16%29%29%2F%28cross%285%29%2Asqrt%282x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel out the common terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Asqrt%288x%5E16%29%29%2F%28sqrt%282x%5E4%29%29 ALIGN=MIDDLE  ALT=\"%283%2Asqrt%288x%5E16%29%29%2F%28sqrt%282x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%28%288x%5E16%29%2F%282x%5E4%29%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%28%288x%5E16%29%2F%282x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%28%284%2A2x%5E16%29%2F%282x%5E4%29%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%28%284%2A2x%5E16%29%2F%282x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 8 into 4*2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%28%284%2Across%282%29x%5E16%29%2F%28cross%282%29x%5E4%29%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%28%284%2Across%282%29x%5E16%29%2F%28cross%282%29x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel out the common terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%28%284x%5E16%29%2F%28x%5E4%29%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%28%284x%5E16%29%2F%28x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%284x%5E%2816-4%29%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%284x%5E%2816-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide the variable terms by subtracting the exponents.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%284x%5E12%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%284x%5E12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%284%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%284%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E12 ALIGN=MIDDLE  ALT=\"x%5E12\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > note: there are 6 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%284%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the square root using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28A%2AB%29=sqrt%28A%29%2Asqrt%28B%29 ALIGN=MIDDLE  ALT=\"sqrt%28A%2AB%29=sqrt%28A%29%2Asqrt%28B%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=3%2A2%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   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=3%2A2%2Ax%2Ax%2Ax%2Ax%2Ax%2Ax ALIGN=MIDDLE  ALT=\"3%2A2%2Ax%2Ax%2Ax%2Ax%2Ax%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   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=6x%5E6 ALIGN=MIDDLE  ALT=\"6x%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%29 ALIGN=MIDDLE  ALT=\"%2815%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E6 ALIGN=MIDDLE  ALT=\"6x%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%29=6x%5E6 ALIGN=MIDDLE  ALT=\"%2815%2Asqrt%288x%5E16%29%29%2F%285%2Asqrt%282x%5E4%29%29=6x%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Edit: The other poster is correct in saying that \"x\" doesn't need to be non-negative, BUT recall that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%29=abs%28x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%29=abs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we assume that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie it is NOT negative), then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%29=x ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > which simplifies things greatly. If \"x\" can be any number, then you must include the absolute value. </SPAN>\n" );
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