document.write( "Question 1155666: [3 + 2√(2)]ⁿ + [3 - 2√(2)]ⁿ = 34 \n" ); document.write( "

Algebra.Com's Answer #778284 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "Notice that  =  = 9 - 8 = 1.\r\n" );
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document.write( "So, if we introduce new variables  x =   and  y = , then we have this system of 2 equations in 2 unknowns\r\n" );
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document.write( "    x + y = 34\r\n" );
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document.write( "    xy    = 1\r\n" );
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document.write( "It leads to the equation in one unknown\r\n" );
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document.write( "    x +  = 34\r\n" );
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document.write( "     = 0\r\n" );
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document.write( "     =  = \r\n" );
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document.write( "So,   =  = 33.97056,\r\n" );
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document.write( "      =  = 0.029437.\r\n" );
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document.write( "Therefore,  if x = 33.97056, then   = 33.97056;  hence  n =  = 2.\r\n" );
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document.write( "            if x = 0.029434, then   = 0.029434;  hence  n =  = -2.\r\n" );
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document.write( "CHECK.  I will check for n = 2 only.\r\n" );
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document.write( "         +  = 3^2 + 8 + 3^2 + 8 = 18 + 16 = 34.   ! Precisely correct !\r\n" );
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document.write( "ANSWER.  n = 2 and/or -2.\r\n" );
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