jmc

Squaring the square root of any number gives that number back. Therefore $${\left(\sqrt{(\sqrt{3}{)}^3}\right)}^4={\left({\left(\sqrt{(\sqrt{3}{)}^3}\right)}^2\right)}^2={\left((\sqrt{3}{)}^3\right)}^2=(\sqrt{3}{)}^6.$$ Again, squaring the square root gives the original number back so $$(\sqrt{3}{)}^6={\left((\sqrt{3}{)}^2\right)}^3=3^3=\boxed{27}.$$