document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=220679>Question 220679</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how to solve radical 2 without using the calculator.\r<BR>\n" );
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document.write( " square root of the cube root of the square root of the cube root of x = x^ to  what power? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #165647 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=165647\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A) how to solve radical 2 without using the calculator<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an irrational number. It cannot be calculated exactly by humans or calculators. Since you do not say how many decimal places you are asked to find for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I'll provide a procedure that is easy to explain:<ol><li>Make a guess.</li><li>Divide the number whose square root you are trying to find by your current guess.</li><li>Make a new guess which is between your previous guess and the quotient you just found.</li><li>Repeat steps 2 and 3 to any desired level of accuracy. The further you go, the harder the long division becomes.</li></ol><BR>\n" );
document.write( "To get you started with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "1. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E2+=+1 ALIGN=MIDDLE  ALT=\"1%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be somewhere between 1 and 2. So we'll start with a guess of 1.5.<BR>\n" );
document.write( "2. Divide 2 by 1.5: 2/1.5 = 1.33333<BR>\n" );
document.write( "3. Make a new guess between 1.5 (the previous guess) and 1.33333 (the quotient). Let's try 1.4.<BR>\n" );
document.write( "4. Repeating step 2 with a current guess of 1.4: 2/1.4 = 1.4285<BR>\n" );
document.write( "5. Repeating step 3: Make a new guess between 1.4 (the previous guess) and 1.4285 (the quotient).<BR>\n" );
document.write( "etc. As you continue, the guesses and quotients will get closer and closer (but never equal!!) so your next guess will require more and more decimal places the get in between them. Stop as soon as you have enough decimal places.<br><BR>\n" );
document.write( "B) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28root%283%2C+sqrt%28root%283%2C+x%29%29%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28root%283%2C+sqrt%28root%283%2C+x%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are somethings we need to understand in order to figure this out:<ul><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28n%2C+x%29+=+x%5E%281%2Fn%29 ALIGN=MIDDLE  ALT=\"root%28n%2C+x%29+=+x%5E%281%2Fn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29+=+root%282%2C+x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29+=+root%282%2C+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5Eb%29%5Ec+=+a%5E%28b%2Ac%29 ALIGN=MIDDLE  ALT=\"%28a%5Eb%29%5Ec+=+a%5E%28b%2Ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li></ul><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28root%283%2C+sqrt%28root%283%2C+x%29%29%29%29+=+%28%28%28x%5E%281%2F3%29%29%5E%281%2F2%29%29%5E%281%2F3%29%29%5E%281%2F2%29+=+x%5E%28%281%2F2%29%2A%281%2F3%29%2A%281%2F2%29%2A%281%2F3%29%29+=+x%5E%281%2F36%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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