document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=631354>Question 631354</A>: <I><SPAN STYLE=\"word-break: break-all;\"> HELLO,<BR>\n" );
document.write( "If anyone answer my questions i will be truly grateful<BR>\n" );
document.write( "Find the square root of: 640.09<BR>\n" );
document.write( "Find the square root of: 9682.56 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #397661 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5396)</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=397661\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume that a pencil and paper calculation is required, because a calculator would give you the answer easily.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "If the square roots are going to be rational numbers, the way to go is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28640.09%29=sqrt%2864009%2F100%29=sqrt%2864009%29%2Fsqrt%28100%29=sqrt%2864009%29%2F10 ALIGN=MIDDLE  ALT=\"sqrt%28640.09%29=sqrt%2864009%2F100%29=sqrt%2864009%29%2Fsqrt%28100%29=sqrt%2864009%29%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%289682.56%29=sqrt%28968256%2F100%29=sqrt%28968256%29%2Fsqrt%28100%29=sqrt%28968256%29%2F10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the square roots are going to be rational numbers, 64009 and 968256 must be perfect squares.<BR>\n" );
document.write( "From there, I would try to factor the numbers in the square roots.<BR>\n" );
document.write( "A complete prime factorization is not needed, but the idea is similar.<BR>\n" );
document.write( " <BR>\n" );
document.write( "64009 is not divisible by 2 (it is not even) or 3 (digits do not add to a multiple of 3).<BR>\n" );
document.write( "It is not divisible by 5 (does not end in 0 or 5) or by 7 (I tried dividing).<BR>\n" );
document.write( "It is divisible by 11, because the sums of alternate digits differ by 11:<BR>\n" );
document.write( "(6+0+9)-(4+0)=15-4=11<BR>\n" );
document.write( "If is is a perfect square, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%5E2 ALIGN=MIDDLE  ALT=\"11%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be a factor, so I can divide by 11 twice.<BR>\n" );
document.write( "I did, and I got 64009/11=5819 and 5819/11=529.<BR>\n" );
document.write( "The number 529 sounded familiar, as if I had known it as a perfect square.<BR>\n" );
document.write( "Since 529 ends in 9, it must be the square of a number that ends in 3 or 7.<BR>\n" );
document.write( "That number be smaller than 25, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%5E2=625%3E529 ALIGN=MIDDLE  ALT=\"25%5E2=625%3E529\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but larger than 20 because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%5E2=400%3C529 ALIGN=MIDDLE  ALT=\"20%5E2=400%3C529\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I tried 23 and found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%5E2=529 ALIGN=MIDDLE  ALT=\"23%5E2=529\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64009=11%2A11%2A23%5E2=11%5E2%2A23%5E2=%2811%2A23%29%5E2=253%5E2 ALIGN=MIDDLE  ALT=\"64009=11%2A11%2A23%5E2=11%5E2%2A23%5E2=%2811%2A23%29%5E2=253%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2864009%29=253 ALIGN=MIDDLE  ALT=\"sqrt%2864009%29=253\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28640.09%29=sqrt%2864009%29%2F10=253%2F10=25.3 ALIGN=MIDDLE  ALT=\"sqrt%28640.09%29=sqrt%2864009%29%2F10=253%2F10=25.3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "968256 is divisible by 2, and if it is a perfect square it must be divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2=4 ALIGN=MIDDLE  ALT=\"2%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So I divided by 4 once to get 968256/4=242064, and again to get 242064/4=60516,<BR>\n" );
document.write( "and a third time to get 60516/4=15129, that is not divisible by 2 any more.<BR>\n" );
document.write( "15129 is divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2=9 ALIGN=MIDDLE  ALT=\"3%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and I divided 15129/9=1681.<BR>\n" );
document.write( "If 1681 going to be a perfect square, it must be the square of a number that ends in 1 or 9, and is a bit larger than 40, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%5E2=1600%3C1681 ALIGN=MIDDLE  ALT=\"40%5E2=1600%3C1681\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "I tried 41 and found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=41%5E2=1681 ALIGN=MIDDLE  ALT=\"41%5E2=1681\"   DISABLED_style= \"max-width:90%; height:auto;\" > so<BR>\n" );
document.write( "41^2*9*4*4*4=41^2*3^2*64=41^2*3^2*8^2=(41*3*8)^2=984^2=968256 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28968256%29=984 ALIGN=MIDDLE  ALT=\"sqrt%28968256%29=984\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%289682.56%29=sqrt%28968256%29%2F10=984%2F10=98.4 ALIGN=MIDDLE  ALT=\"sqrt%289682.56%29=sqrt%28968256%29%2F10=984%2F10=98.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "There is a general way to find square roots, that even allows you to get approximate values for irrational squre roots. It can be worked into a procedure sort of like long division, which I was taught in school many, many years ago. I hope that is not what was expected from you in the era of smartphones and tablet computers. </SPAN>\n" );
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