document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209711>Question 1209711</A>: <I><SPAN STYLE=\"word-break: break-all;\"> [x-(1/x)]^(1/x) + [1-(1/x)] ^(1/x) = x  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849937 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849937\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to approach this problem:\r<BR>\n" );
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document.write( "1. **Consider x = 1:**\r<BR>\n" );
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document.write( "If x = 1, the equation becomes:\r<BR>\n" );
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document.write( "[1 - (1/1)]^(1/1) + [1 - (1/1)]^(1/1) = 1\r<BR>\n" );
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document.write( "[1 - 1]^1 + [1 - 1]^1 = 1\r<BR>\n" );
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document.write( "0^1 + 0^1 = 1\r<BR>\n" );
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document.write( "0 + 0 = 1\r<BR>\n" );
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document.write( "0 = 1\r<BR>\n" );
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document.write( "This is not true, so x = 1 is not a solution.\r<BR>\n" );
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document.write( "2. **Rewrite the equation:**\r<BR>\n" );
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document.write( "Let's rewrite the given equation:\r<BR>\n" );
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document.write( "(x - 1)/x ]^(1/x) + [(x - 1)/x]^(1/x) = x\r<BR>\n" );
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document.write( "2 * [(x - 1)/x]^(1/x) = x\r<BR>\n" );
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document.write( "3. **Consider the case where x = 2:**\r<BR>\n" );
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document.write( "2 * [(2 - 1)/2]^(1/2) = 2\r<BR>\n" );
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document.write( "2 * (1/2)^(1/2) = 2\r<BR>\n" );
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document.write( "2 * (1/√2) = 2\r<BR>\n" );
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document.write( "√2 = 2\r<BR>\n" );
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document.write( "This is not true, so x=2 is not a solution.\r<BR>\n" );
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document.write( "4. **Consider the case where x = 4:**\r<BR>\n" );
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document.write( "2 * [(4 - 1)/4]^(1/4) = 4\r<BR>\n" );
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document.write( "2 * (3/4)^(1/4) = 4\r<BR>\n" );
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document.write( "(3/4)^(1/4) = 2\r<BR>\n" );
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document.write( "Raising both sides to the power of 4:\r<BR>\n" );
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document.write( "3/4 = 2^4\r<BR>\n" );
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document.write( "3/4 = 16\r<BR>\n" );
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document.write( "This is not true, so x=4 is not a solution.\r<BR>\n" );
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document.write( "5. **A more general approach is difficult:** This equation is a transcendental equation (it mixes polynomials and non-algebraic functions).  There's no simple algebraic method to solve it.  Numerical methods (like the Newton-Raphson method) or a computer algebra system are usually required.\r<BR>\n" );
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document.write( "6. **Graphical Approach:**  The best way to find a solution is to graph both sides of the equation, y = 2 * [(x - 1)/x]^(1/x) and y = x, and look for the intersection point.  This will give you an approximate solution.\r<BR>\n" );
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document.write( "7. **Numerical Solution:**  Using a numerical solver (like Wolfram Alpha or a calculator with a solve function), we can find an approximate solution. It will be a value greater than 1.\r<BR>\n" );
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document.write( "It is difficult to find a closed-form solution. The most practical approach is to use a numerical solver or a graphing calculator to approximate the solution.<BR>\n" );
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