The imaginary part of 3 + 2 - 54 1 2 - 3 - 2 - 54 1 2 ,can be

$|3+2\sqrt{-54}|=\sqrt{9+216}=15$

${\left(3+2\sqrt{-54}\right)}^{1/2}=\pm \left(\sqrt{\frac{15+3}{2}}+i\sqrt{\frac{15-3}{2}}\right)$

$=\pm \left(3+i\sqrt{6}\right)$ 

Also ${\left(3-2\sqrt{-54}\right)}^{1/2}=\pm \left(3-i\sqrt{6}\right)$ 

Hence $\left\{{\left(3+2\sqrt{-54}\right)}^{1/2}-{\left(3-2\sqrt{-54}\right)}^{1/2}\right\}$ 

$=\pm 2\sqrt{6}i$ or $\pm 6$ 

Hence, imaginary part $=\pm 2\sqrt{6}$