Solve in R : $\displaystyle \left( {x^x } \right)^2 = 8$
$\displaystyle x^{x}= \sqrt{8} = 2^{3/2} $
$\displaystyle e^{\ln x \ e^{\ln x}} = 2^{3/2} $
$\displaystyle \ln (e^{\ln x \ e^{\ln x}}) = \ln (2^{3/2})$
$\displaystyle \ln x e^{\ln x} =\frac{3}{2} \ln 2$
$\displaystyle \ln x = \frac{\frac{3}{2}\ln 2}{e^{\ln x}} = W\Big(\frac{3}{2} \ln 2\Big)$
where W is the Lambert W function
so $\displaystyle x = e^{W\Big(\frac{3}{2} \ln 2\Big)} \approx 1.788 $