Complex exercise

Exercise:

What is the value of $\sqrt{{{\left( \sqrt{-1} \right)}^{\left( \sqrt{-1} \right)}}}=??$

Solution: we Know that ${{i}^{2}}=-1\Leftrightarrow \sqrt{-1}=\sqrt{{{i}^{2}}}=i$

So $\sqrt{{{\left( \sqrt{-1} \right)}^{\left( \sqrt{-1} \right)}}}=\sqrt{{{i}^{i}}}$

But $i={{e}^{i\frac{\pi }{2}}}\Rightarrow {{i}^{i}}={{e}^{\left( i\frac{\pi }{2} \right)i}}={{e}^{-\frac{\pi }{2}}}=\frac{1}{{{e}^{\frac{\pi }{2}}}}=\frac{1}{\sqrt{{{e}^{\pi }}}}$

So $\sqrt{{{i}^{i}}}=\sqrt{\frac{1}{\sqrt{{{e}^{\pi }}}}}=\frac{1}{\sqrt{\sqrt{{{e}^{\pi }}}}}=\frac{1}{{{e}^{\frac{\pi }{4}}}}={{e}^{-\frac{\pi }{4}}}=0.45593812776599624$