The imaginary part of

$(3+2 \sqrt{-54})=3+2 \times 3 \times \sqrt{6} \mathrm{i}$

$=(3+\sqrt{6} i)^{2}$

$(3-2 \sqrt{54})=(3-\sqrt{6} i)^{2}$

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

$=\pm(3+\sqrt{6} \mathrm{i}) \pm(3-\sqrt{6} \mathrm{i})$

$=6,-6,2 \sqrt{6} i,-2 \sqrt{6} i$