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  4. The imaginary part of (3+2 √-54)1 / 2-(3-2 √-54)1 / 2 can be

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Q. The imaginary part of
$(3+2 \sqrt{-54})^{1 / 2}-(3-2 \sqrt{-54})^{1 / 2}$ can be

JEE MainJEE Main 2020Complex Numbers and Quadratic Equations

Solution:

$(3+2 \sqrt{-54})=3+2 \times 3 \times \sqrt{6} i$
$=(3+\sqrt{6} i )^{2}$
$(3-2 \sqrt{54})=(3-\sqrt{6} i)^{2}$
$(3+2 \sqrt{-54})^{1 / 2}+(3-2 \sqrt{-54})^{1 / 2}$
$=\pm(3+\sqrt{6} i) \pm(3-\sqrt{6} i)$
$=6,-6,2 \sqrt{6} i ,-2 \sqrt{6} i ,$ (four possible answer)