Question

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

• A. $\sqrt{6}$
• B. $-2\sqrt{6}$
• C. $6$
• D. $-\sqrt{6}$

Answer 1

The correct option is B $-2\sqrt{6}$

${(3+2i\sqrt{54})}^{1/2}-{(3-2i\sqrt{54})}^{1/2}$
$={(9+6i^2+2\cdot 3\cdot i\sqrt{6})}^{1/2}-{(9+6i^2-2\cdot 3\cdot i\sqrt{6})}^{1/2}$
$={\left({(3+i\sqrt{6})}^2\right)}^{1/2}-{\left({(3-i\sqrt{6})}^2\right)}^{1/2}$
$=\pm (3+i\sqrt{6})\mp (3-i\sqrt{6})=\pm 2\sqrt{6}i$