document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=825831>Question 825831</A>: <I><SPAN STYLE=\"word-break: break-all;\"> evaluate without using a calculator, using binomial theorem,<BR>\n" );
document.write( "1) 4th root (624)<BR>\n" );
document.write( "2) 1/(1+x)^1/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #497649 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=497649\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) I am not sure exactly what you are expected to do to evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2C624%29 ALIGN=MIDDLE  ALT=\"root%284%2C624%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > without using a calculator, using binomial theorem, but this is what I would do.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E4=256 ALIGN=MIDDLE  ALT=\"4%5E4=256\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E4=625 ALIGN=MIDDLE  ALT=\"5%5E4=625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since 624 is so close to 625, i would expect <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2C624%29 ALIGN=MIDDLE  ALT=\"root%284%2C624%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be very close to 5, so let's try 4.9.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.9%5E4=%285-0.1%29%5E4=g%5E4-4%2A5%5E3%2A0.1%2B6%2A5%5E2%2A0.1%5E2-4%2A5%2A0.1%5E3%2B0.1%5E4=625-4%2A125%2A0.1%2B6%2A25%2A0.01-4%2A5%2A.001%2B.0001=625-50%2B1.5-.02%2B.0001=576.4801 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since 624 is much closer to 625 than to 576.48, I would expect <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2C624%29 ALIGN=MIDDLE  ALT=\"root%284%2C624%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be much closer to 5 than to 4.9, so I would try 4.99.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.99%5E4=%285-0.01%29%5E4=g%5E4-4%2A5%5E3%2A0.01%2B6%2A5%5E2%2A0.01%5E2-4%2A5%2A0.01%5E3%2B0.01%5E4=625-4%2A125%2A0.01%2B6%2A25%2A0.0001-4%2A5%2A0.000001%2B0.00000001 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= about<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=625-5%2B0.015=620.015 ALIGN=MIDDLE  ALT=\"625-5%2B0.015=620.015\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point, knowing that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.99%5E4=about620.015 ALIGN=MIDDLE  ALT=\"4.99%5E4=about620.015\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.00%5E4=625.000 ALIGN=MIDDLE  ALT=\"5.00%5E4=625.000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I would estimate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2C624%29=4.998 ALIGN=MIDDLE  ALT=\"root%284%2C624%29=4.998\"   DISABLED_style= \"max-width:90%; height:auto;\" > by assuming that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=root%284%2Cx%29 ALIGN=MIDDLE  ALT=\"f%28x%29=root%284%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is approximately linear between 4.990 and 5.000.<BR>\n" );
document.write( " <BR>\n" );
document.write( "2) There is nothing that can be evaluated in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281%2Bx%29%5E1%2F2=1%2Fsqrt%281%2Bx%29 ALIGN=MIDDLE  ALT=\"1%2F%281%2Bx%29%5E1%2F2=1%2Fsqrt%281%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , since we do not have a value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That expression can be rationalized as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281%2Bx%29%5E1%2F2=1%2Fsqrt%281%2Bx%29=%281%2Fsqrt%281%2Bx%29%29%2A%28sqrt%281%2Bx%29%2Fsqrt%281%2Bx%29%29=1%2Asqrt%281%2Bx%29%2F%28sqrt%281%2Bx%29%2Asqrt%281%2Bx%29%29=sqrt%281%2Bx%29%2F%28sqrt%281%2Bx%29%29%5E2=sqrt%281%2Bx%29%2F%281%2Bx%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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