# The imaginary part of (3 + 2√−54)^1/2 − (3 − 2√−54)^1/2 can be

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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

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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 $=\pm2\sqrt6$