document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209755>Question 1209755</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If x = [(a + √(a² - 1)]^(2mn/(m-n)), <BR>\n" );
document.write( "Then [x^(1/m) + x^(1/n)]²/[x^[(1/m)+(1/n)]] = 1444,<BR>\n" );
document.write( "find a.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850157 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=850157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step:\r<BR>\n" );
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document.write( "**1. Simplify the Expression:**\r<BR>\n" );
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document.write( "Let's simplify the expression:\r<BR>\n" );
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document.write( "[x^(1/m) + x^(1/n)]² / x^[(1/m) + (1/n)]\r<BR>\n" );
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document.write( "= [x^(1/m) + x^(1/n)]² / x^[(m+n)/mn]\r<BR>\n" );
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document.write( "= [x^(1/m)]² + 2x^(1/m)x^(1/n) + [x^(1/n)]² / x^[(m+n)/mn]\r<BR>\n" );
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document.write( "= [x^(2/m) + 2x^((1/m)+(1/n)) + x^(2/n)] / x^[(m+n)/mn]\r<BR>\n" );
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document.write( "= x^(2/m) / x^[(m+n)/mn] + 2x^((1/m)+(1/n)) / x^[(m+n)/mn] + x^(2/n) / x^[(m+n)/mn]\r<BR>\n" );
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document.write( "= x^(2/m - (m+n)/mn) + 2x^((1/m)+(1/n) - (m+n)/mn) + x^(2/n - (m+n)/mn)\r<BR>\n" );
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document.write( "= x^((2n - m - n)/mn) + 2x^((mn(1/m+1/n) - m - n)/mn) + x^((2m - m - n)/mn)\r<BR>\n" );
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document.write( "= x^((n - m)/mn) + 2x^((m+n - m - n)/mn) + x^((m - n)/mn)\r<BR>\n" );
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document.write( "= x^((n - m)/mn) + 2x^0 + x^((m - n)/mn)\r<BR>\n" );
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document.write( "= x^((n - m)/mn) + 2 + x^((m - n)/mn)\r<BR>\n" );
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document.write( "= x^((n - m)/mn) + 2 + x^(-(n - m)/mn)\r<BR>\n" );
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document.write( "Let y = (n - m)/mn. Then the expression becomes:\r<BR>\n" );
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document.write( "x^y + 2 + x^(-y)\r<BR>\n" );
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document.write( "**2. Substitute the Value of x:**\r<BR>\n" );
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document.write( "Given x = [(a + √(a² - 1)]^(2mn/(m-n)), we have:\r<BR>\n" );
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document.write( "x^y = [(a + √(a² - 1)]^(2mn/(m-n)) * (n-m)/mn = (a + √(a² - 1))^(-2)\r<BR>\n" );
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document.write( "x^(-y) = [(a + √(a² - 1)]^(2mn/(m-n)) * (m-n)/mn = (a + √(a² - 1))^(2)\r<BR>\n" );
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document.write( "Therefore, the expression becomes:\r<BR>\n" );
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document.write( "(a + √(a² - 1))^(-2) + 2 + (a + √(a² - 1))^(2) = 1444\r<BR>\n" );
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document.write( "**3. Simplify Further:**\r<BR>\n" );
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document.write( "Let z = a + √(a² - 1). Then:\r<BR>\n" );
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document.write( "1/z² + 2 + z² = 1444\r<BR>\n" );
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document.write( "z² + 1/z² = 1442\r<BR>\n" );
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document.write( "(z + 1/z)² = z² + 2 + 1/z² = 1444\r<BR>\n" );
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document.write( "z + 1/z = ±38\r<BR>\n" );
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document.write( "**4. Solve for z:**\r<BR>\n" );
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document.write( "z + 1/z = 38 (we can ignore the negative solution as z is positive)\r<BR>\n" );
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document.write( "z² - 38z + 1 = 0\r<BR>\n" );
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document.write( "Using the quadratic formula:\r<BR>\n" );
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document.write( "z = [38 ± √(38² - 4)] / 2\r<BR>\n" );
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document.write( "z = [38 ± √1440] / 2\r<BR>\n" );
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document.write( "z = 19 ± √360\r<BR>\n" );
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document.write( "z = 19 ± 6√10\r<BR>\n" );
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document.write( "Since z = a + √(a² - 1), we have:\r<BR>\n" );
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document.write( "a + √(a² - 1) = 19 + 6√10\r<BR>\n" );
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document.write( "a + √(a² - 1) = (3+√10)²\r<BR>\n" );
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document.write( "a + √(a² - 1) = 3² + 2*3*√10 + (√10)²\r<BR>\n" );
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document.write( "a + √(a² - 1) = 9 + 6√10 + 10\r<BR>\n" );
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document.write( "a + √(a² - 1) = 19 + 6√10\r<BR>\n" );
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document.write( "Therefore a = 19.\r<BR>\n" );
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document.write( "**5. Verification:**\r<BR>\n" );
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document.write( "If a = 19, then √(a² - 1) = √(19² - 1) = √360 = 6√10\r<BR>\n" );
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document.write( "a + √(a² - 1) = 19 + 6√10\r<BR>\n" );
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document.write( "**Final Answer:**\r<BR>\n" );
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document.write( "a = 19<BR>\n" );
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