document.write( "Question 1209712: If (x√x)^(1/x) = 2,
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Algebra.Com's Answer #849939 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "If (x√x)^(1/x) = 2, find x.
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document.write( "The domain of the given equation is the set of positive real numbers x > 0.\r\n" );
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document.write( "Let's analyze the function in the left side.\r\n" );
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document.write( "Take logarithm of it:\r\n" );
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document.write( "    \"ln%28%28x%2Asqrt%28x%29%29%5E%281%2Fx%29%29\" = \"%283%2F2%29%2A%28ln%28x%29%2Fx%29\".\r\n" );
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document.write( "It is well known fact that function  \"ln%28x%29%2Fx%29\"  is a limited function and has \r\n" );
document.write( "the maximum at  x = e = 2.718  (the base of natural logarithms).\r\n" );
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document.write( "     It can be easy verified by the standard Calculus procedure\r\n" );
document.write( "     taking the derivative and equating it to zero.\r\n" );
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document.write( "So, let's evaluate the function in the left side of the original equation.\r\n" );
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document.write( "It is  \"%28e%2Asqrt%28e%29%29%5E%281%2Fe%29\" = \"%282.718%2Asqrt%282.718%29%29%5E%281%2F2.718%29\" = 1.736409.\r\n" );
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document.write( "In any case, the left side of the original equation is ALWAYS less than 2,\r\n" );
document.write( "so, the given equation HAS NO solutions in the domain x > 0.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The plot of the function in left side of the original equation is shown under this lin\r
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\n" ); document.write( "\n" ); document.write( "https://www.desmos.com/calculator/sl4wgvfqse\r
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\n" ); document.write( "\n" ); document.write( "https://www.desmos.com/calculator/sl4wgvfqse\r
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\n" ); document.write( "\n" ); document.write( "It shows visually that the left side is  ALWAYS  less than  2,  for all positive real values of  x.\r
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\n" ); document.write( "\n" ); document.write( "The solution in the post by  @CPhill is  INCORRECT.\r
\n" ); document.write( "\n" ); document.write( "His answer,  giving the root of the original equation about  3.5,  is  WRONG.\r
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\n" ); document.write( "\n" ); document.write( "I checked the value of the left side function at  x = 3.5.  It is\r
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\n" ); document.write( "         \"%283.5%2Asqrt%283.5%29%29%5E%281%2F3.5%29\" ~ 1.70693,\r
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\n" ); document.write( "\n" ); document.write( "which is  LESS  than  2.\r
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\n" ); document.write( "\n" ); document.write( "So,  ignore the post by  @CPhill,  since it is  FULL  of  FATAL  ERRORS.\r
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