The imaginary part of (3 + 2 - 54 )(1)/(2) - (3 - 2 - 54 )(1)/(2)...

| 3 + 2\sqrt{- 54} | = \sqrt{9 + 216} = 15$</p><p>${\left(3 + 2\sqrt{- 54}\right)}^{1 / 2} = \pm\left(\sqrt{\dfrac{15 + 3}{2}} + i\sqrt{\dfrac{15 - 3}{2}}\right)$</p><p>$= \pm \left(3 + i\sqrt{6}\right)

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

$= \pm 2\sqrt{6}i$ or $\pm 6$

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