The imaginary part of

Question:

The imaginary part of

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

 

  1. $-2 \sqrt{6}$

  2. 6

  3. $\sqrt{6}$

  4. $-\sqrt{6}$


Correct Option: 1

Solution:

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

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

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

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

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

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

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