Question 1209712
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If (x√x)^(1/x) = 2, find x. 
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<pre>
The domain of the given equation is the set of positive real numbers x > 0.


Let's analyze the function in the left side.

Take logarithm of it:

    {{{ln((x*sqrt(x))^(1/x))}}} = {{{(3/2)*(ln(x)/x)}}}.


It is well known fact that function  {{{ln(x)/x)}}}  is a limited function and has 
the maximum at  x = e = 2.718  (the base of natural logarithms).


     It can be easy verified by the standard Calculus procedure
     taking the derivative and equating it to zero.


So, let's evaluate the function in the left side of the original equation.

It is  {{{(e*sqrt(e))^(1/e)}}} = {{{(2.718*sqrt(2.718))^(1/2.718)}}} = 1.736409.


In any case, the left side of the original equation is ALWAYS less than 2,
so, the given equation HAS NO solutions in the domain x > 0.
</pre>

Solved.


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The plot of the function in left side of the original equation is shown under this lin


<A HREF=https://www.desmos.com/calculator/sl4wgvfqse>https://www.desmos.com/calculator/sl4wgvfqse</A>


https://www.desmos.com/calculator/sl4wgvfqse


It shows visually that the left side is &nbsp;ALWAYS &nbsp;less than &nbsp;2, &nbsp;for all positive real values of &nbsp;x.



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The solution in the post by &nbsp;@CPhill is &nbsp;INCORRECT.

His answer, &nbsp;giving the root of the original equation about &nbsp;3.5, &nbsp;is &nbsp;WRONG.


I checked the value of the left side function at &nbsp;x = 3.5.  &nbsp;It is

   
    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{(3.5*sqrt(3.5))^(1/3.5)}}} ~ 1.70693,


which is &nbsp;LESS &nbsp;than &nbsp;2.


So, &nbsp;ignore the post by &nbsp;@CPhill, &nbsp;since it is &nbsp;FULL &nbsp;of &nbsp;FATAL &nbsp;ERRORS.