Question

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

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

Answer 1

The correct option is B

$-2\sqrt{6}$

Find the imaginary part of the given expression

Given expression, ${\left(3+2\sqrt{-54}\right)}^{\frac{1}{2}}-{\left(3-2\sqrt{-54}\right)}^{\frac{1}{2}}$

Upon simplifying we get,

${\left(3+2\sqrt{-54}\right)}^{\frac{1}{2}}-{\left(3-2\sqrt{-54}\right)}^{\frac{1}{2}}$ $=$ ${\left(3+2i\sqrt{54}\right)}^{\frac{1}{2}}-{\left(3-2i\sqrt{54}\right)}^{\frac{1}{2}}$

$={\left(9+6i^2+2\times 3\times i\sqrt{6}\right)}^{\frac{1}{2}}-{\left(9+6i^2-2\times 3\times i\sqrt{6}\right)}^{\frac{1}{2}}$

$={\left({\left(3+i\sqrt{6}\right)}^2\right)}^{\frac{1}{2}}-{\left({\left(3-i\sqrt{6}\right)}^2\right)}^{\frac{1}{2}}$

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

$\therefore$${\left(3+2\sqrt{-54}\right)}^{\frac{1}{2}}-{\left(3-2\sqrt{-54}\right)}^{\frac{1}{2}}$ $=6,-6,2\sqrt{6}i,-2\sqrt{6}i$

From these answers and given options we can see that $-2\sqrt{6}$ is the imaginary part of the given expression

Hence, option $\left(B\right)$ $-2\sqrt{6}$, is the correct answer.