document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=107071>Question 107071</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi - In a recent demonstration, somebody went from this<BR>\n" );
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document.write( "(square root of (1 + x-squared))/x\r<BR>\n" );
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document.write( "to this<BR>\n" );
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document.write( "square root of (1 + 1/x-squared)\r<BR>\n" );
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document.write( "They're both obviously equivalent, as both result in the same answer for values of x.   Can you help me to understand how this manipulation was performed, and how these two forms relate?  Thanks!\r<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 #77878 by </B><A HREF=/tutors/aboutme.mpl?userid=HyperBrain><B>HyperBrain(694)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=HyperBrain><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=77878\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "THIS IS INTERESTING!!! Thanks! I love this type of problem.\r<BR>\n" );
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document.write( "Note that the equation above is the same as\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29%2F%281%29 ALIGN=MIDDLE  ALT=\"sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "By cross multiplication,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28sqrt%281%2B%281%2Fx%5E2%29%29%29=1%28sqrt%281%2Bx%5E2%29%29 ALIGN=MIDDLE  ALT=\"x%28sqrt%281%2B%281%2Fx%5E2%29%29%29=1%28sqrt%281%2Bx%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28sqrt%281%2B%281%2Fx%5E2%29%29%29=sqrt%281%2Bx%5E2%29 ALIGN=MIDDLE  ALT=\"x%28sqrt%281%2B%281%2Fx%5E2%29%29%29=sqrt%281%2Bx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you square both sides,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%28sqrt%281%2B%281%2Fx%5E2%29%29%29%29%5E2=%28sqrt%281%2Bx%5E2%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%28sqrt%281%2B%281%2Fx%5E2%29%29%29%29%5E2=%28sqrt%281%2Bx%5E2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%281%2B1%2Fx%5E2%29%29=%281%2Bx%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E2%281%2B1%2Fx%5E2%29%29=%281%2Bx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bx%5E2 ALIGN=MIDDLE  ALT=\"1%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%281%29+%2B+x%5E2%281%2Fx%5E2%29=1%2Bx%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%281%29+%2B+x%5E2%281%2Fx%5E2%29=1%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B+%28x%5E2%2Fx%5E2%29=1%2Bx%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B+%28x%5E2%2Fx%5E2%29=1%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B1=1%2Bx%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B1=1%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >-------------we're getting close! watch the next move or<BR>\n" );
document.write( "                              maybe you can do it. hint:it IS easy\r<BR>\n" );
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document.write( "The final move....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B1=x%5E2%2B1 ALIGN=MIDDLE  ALT=\"x%5E2%2B1=x%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--------------can you guess what I did? From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bx%5E2 ALIGN=MIDDLE  ALT=\"1%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I<BR>\n" );
document.write( "                               just interchanged <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%281%2Bx%5E2%29%2Fx=sqrt%281%2B%281%2Fx%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is proven to be true.\r<BR>\n" );
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document.write( "Power up,<BR>\n" );
document.write( "HyperBrain! </SPAN>\n" );
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