Solve in R :
The analytic solution of x^sqrt(x)=1/2 is
x=exp(2*W(-ln(2)/2)) where W is the LambertW function.
The LambertW function is multi-valuated. Numerical evalation leads to
x=0.25 and x=0.0625
Of course, the values can be found much more simply, by direct evalation.