document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86391>Question 86391</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help me with these questions?<BR>\n" );
document.write( "a) sq. rt. of 12 plus sq.rt. of 108<BR>\n" );
document.write( "b) ^3 sq. rt. of 16 times ^3 sq. rt. of 3<BR>\n" );
document.write( "Thank you so much for your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62500 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=62500\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2812%29%2Bsqrt%28108%29 ALIGN=MIDDLE  ALT=\"sqrt%2812%29%2Bsqrt%28108%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First lets simplify <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2812%29 ALIGN=MIDDLE  ALT=\"sqrt%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2812%29 ALIGN=MIDDLE  ALT=\"sqrt%2812%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" );
document.write( "\n" );
document.write( "So let's list the factors of 12\r<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1, 2, 3, 4, 6,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how 4 is the largest perfect square, so lets break 12 down into 4*3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2A3%29 ALIGN=MIDDLE  ALT=\"sqrt%284%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 12 into 4*3<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%29%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%284%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" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of the perfect square 4 to get 2 <BR>\n" );
document.write( " <BR>\n" );
document.write( "So the expression\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2812%29 ALIGN=MIDDLE  ALT=\"sqrt%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "simplifies to\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Now lets simplify <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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
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" );
document.write( "\n" );
document.write( "So let's list the factors of 108\r<BR>\n" );
document.write( "\n" );
document.write( "Factors:\r<BR>\n" );
document.write( "\n" );
document.write( "1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how 36 is the largest perfect square, so lets break 108 down into 36*3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( " <BR>\n" );
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" );
document.write( " <BR>\n" );
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" );
document.write( " <BR>\n" );
document.write( "So the expression\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "simplifies to\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the expression\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2812%29%2Bsqrt%28108%29 ALIGN=MIDDLE  ALT=\"sqrt%2812%29%2Bsqrt%28108%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "simplifies to \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%283%29%2B6%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%283%29%2B6%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sqrt%283%29 ALIGN=MIDDLE  ALT=\"y=sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%2B6y ALIGN=MIDDLE  ALT=\"2y%2B6y\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now combine like terms\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y ALIGN=MIDDLE  ALT=\"8y\"   DISABLED_style= \"max-width:90%; height:auto;\" >}\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Replace y with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"8%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Evaluate the given expression with a calculator:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2812%29%2Bsqrt%28108%29=13.856406460551 ALIGN=MIDDLE  ALT=\"sqrt%2812%29%2Bsqrt%28108%29=13.856406460551\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Evaluate the simplified expression with a calculator:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Asqrt%283%29=13.856406460551 ALIGN=MIDDLE  ALT=\"8%2Asqrt%283%29=13.856406460551\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since they are equal (to a certain decimal place), this verifies our answer\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C16%29%2Aroot%283%2C3%29 ALIGN=MIDDLE  ALT=\"root%283%2C16%29%2Aroot%283%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the 2 radicands are under the same root value, we can combine them using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2Cx%29%2Aroot%28n%2Cy%29=root%28n%2Cx%2Ay%29 ALIGN=MIDDLE  ALT=\"root%28n%2Cx%29%2Aroot%28n%2Cy%29=root%28n%2Cx%2Ay%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C16%2A3%29 ALIGN=MIDDLE  ALT=\"root%283%2C16%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the cube roots\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C48%29 ALIGN=MIDDLE  ALT=\"root%283%2C48%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C8%2A6%29 ALIGN=MIDDLE  ALT=\"root%283%2C8%2A6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 48 into 8*6. I chose to factor out an 8 since 8 is a perfect cube\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C8%29%2Aroot%283%2C6%29 ALIGN=MIDDLE  ALT=\"root%283%2C8%29%2Aroot%283%2C6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the roots using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2Cx%2Ay%29=root%28n%2Cx%29%2Aroot%28n%2Cy%29 ALIGN=MIDDLE  ALT=\"root%28n%2Cx%2Ay%29=root%28n%2Cx%29%2Aroot%28n%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aroot%283%2C6%29 ALIGN=MIDDLE  ALT=\"2%2Aroot%283%2C6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the cube root of 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Evaluate the given expression with a calculator:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C16%29%2Aroot%283%2C3%29=3.63424118566428 ALIGN=MIDDLE  ALT=\"root%283%2C16%29%2Aroot%283%2C3%29=3.63424118566428\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Evaluate the simplified expression with a calculator:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aroot%283%2C6%29=3.63424118566428 ALIGN=MIDDLE  ALT=\"2%2Aroot%283%2C6%29=3.63424118566428\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since they are equal (to a certain decimal place), this verifies our answer </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );