The imaginary part of $(3+2\sqrt{-54})^{1\over2}-(3-2\sqrt{-54})^{1\over2}$ can be
(1) $\sqrt{-6}$
(2) $\sqrt{6}$
(3) $-2\sqrt{6}$
(4) 6
(1) $\sqrt{-6}$
(2) $\sqrt{6}$
(3) $-2\sqrt{6}$
(4) 6
Ans. (3) $−2\sqrt6$
Sol. $|3+2\sqrt{-54}|=\sqrt{9+216}=15$
$\implies(3+2\sqrt{-54})^{1/2}=\pm\left(\sqrt{15+3\over2}+i\sqrt{15-3\over2}\right)$
$=\pm(3 + i\sqrt6)$
and $(3-2\sqrt{-54})^{1/2}=\pm(3-i\sqrt6)$
Hence $\left\{(3+2\sqrt{-54})^{1/2}-(3-2\sqrt{-54})^{1/2}\right\}$
$=\pm2i\sqrt6\text{ or }\pm6$
Hence imaginary part $=\pm2\sqrt6$