Question 1196903
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Your starting equation is 

    {{{sqrt(x)^x}}} = {{{x^(sqrt(x))}}}.       (1)


One solution is x= 1 (obvious).


Indeed,  left side is {{{sqrt(1)^1}}} = {{{1^1}}} = 1.

         Right side is equal to 1 due to the same reason.


Now I will assume that x =/= 1 and will look for other solutions.



Take logarithm base 10 of both sides of equation (1).  You will get

    {{{x*log((sqrt(x)))}}} = {{{sqrt(x)*log((x))}}}.


Divide both sides by log(x)  (we can do it safely, since we consider  x =/= 1.)
You will get

    {{{sqrt(x)/x}}} = {{{log((sqrt(x)))/log((x))}}},


or, equivalently,

    {{{sqrt(x)/x}}} = {{{1/2}}}.


It implies

    {{{2*sqrt(x)}}} = x

and after squaring both sides,

    4x = {{{x^2}}}.


It implies

    {{{4x - x^2}}} = 0

    x*(4-x) = 0,

    x = 0  or   x= 4.


You can check that the root x= 0 works in the original equation, since then each side is equal to 1.

So, the remaining solutions to the problem are  x= 0  and  x= 4.


<U>ANSWER</U>.  The given equation has three solutions  x= 0,  x= 1,  and x= 4.
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Solved.


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                  To see that x= 0 is the solution, too, look at this plot below.



    {{{ graph(800,400, -0.2, 1.2, -1, 2, sqrt(x)^x, x^sqrt(x) ) }}}


                        Plots y = {{{sqrt(x)^x}}}  (red)  and  y = {{{x^sqrt(x)}}}  (green)
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