document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=983439>Question 983439</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. IF X=7-4RT3, FIND THE VALUE OF RTX+1/RTX?<BR>\n" );
document.write( "2. X=3+2RT2, FIND THE VALUE OF X^3-1/X^3?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #604324 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52800)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=604324\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "1. Let me re-formulate your first request in the correct algebraic form:\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7-4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;find the value &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%28x%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<B>Solution</B>\r<BR>\n" );
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document.write( "We have &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7-4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;Let &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > be &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >: &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Then &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ay ALIGN=MIDDLE  ALT=\"x%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287-4sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%287-4sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2B4sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%287%2B4sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E2 ALIGN=MIDDLE  ALT=\"7%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284sqrt%283%29%29%5E2 ALIGN=MIDDLE  ALT=\"%284sqrt%283%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49 ALIGN=MIDDLE  ALT=\"49\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2A3 ALIGN=MIDDLE  ALT=\"16%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49 ALIGN=MIDDLE  ALT=\"49\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48 ALIGN=MIDDLE  ALT=\"48\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;(I applied here the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%2A%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"%28a-b%29%2A%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+-+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;Are you familiar with it ?)\r<BR>\n" );
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document.write( "Thus we obtain a remarkable formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287-4sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%287-4sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2B4sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%287%2B4sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "It means that the numbers &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7-4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; are reciprocal each to the other: &nbsp;their product is equal to &nbsp;1. \r<BR>\n" );
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document.write( "In other words, &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy ALIGN=MIDDLE  ALT=\"1%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;or &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;i.e. &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now, &nbsp;let us square &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29+%2B+1%2Fsqrt%28x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29+%2B+1%2Fsqrt%28x%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=%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28x%29%2A%281%2Fsqrt%28x%29%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28x%29%2A%281%2Fsqrt%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28sqrt%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"1%2F%28sqrt%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;(I applied here the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+%2B2ab+%2B+b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+%2B2ab+%2B+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;Hope you are familiar with it.) \r<BR>\n" );
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document.write( "Substitute here &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7-4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; and &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; as we found above. \r<BR>\n" );
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document.write( "You can continue this identity as \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28x%29%2A%281%2Fsqrt%28x%29%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28x%29%2A%281%2Fsqrt%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28sqrt%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"1%2F%28sqrt%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7-4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B4sqrt%283%29 ALIGN=MIDDLE  ALT=\"7%2B4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = &nbsp;&nbsp;(now collect the common terms and annihilate the terms with &nbsp;<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" );
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document.write( "= 7 + 7 + 2 = 16.\r<BR>\n" );
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document.write( "Since &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%29+%2B+1%2Fsqrt%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 16, &nbsp;you get &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29+%2B+1%2Fsqrt%28x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29+%2B+1%2Fsqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 4. \r<BR>\n" );
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document.write( "Amazingly, &nbsp;isn't?\r<BR>\n" );
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document.write( "The solution is completed.\r<BR>\n" );
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document.write( "The problem and the solution are like sweet cookies. &nbsp;Thank you &nbsp;(and yours teacher)&nbsp; for giving it. \r<BR>\n" );
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document.write( "2. &nbsp;Regarding the second problem of your request, &nbsp;try to solve it yourself similar to the first one.\r<BR>\n" );
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document.write( "You have &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"3%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Introduce &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"3-2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Check that &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ay ALIGN=MIDDLE  ALT=\"x%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1, &nbsp;&nbsp;i.e. the numbers &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"3%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; and &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"3-2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; are reciprocal &nbsp;&nbsp;and &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3-2sqrt%282%29 ALIGN=MIDDLE  ALT=\"3-2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Based on it, &nbsp;complete the solution yourself.\r<BR>\n" );
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document.write( "Good luck !!!\r<BR>\n" );
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document.write( " </SPAN>\n" );
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