Question

Evaluate : $\sqrt{-8-6\mathrm{i}}$

• A. $1\pm 3\mathrm{i}$
• B. $\pm (1-3\mathrm{i})$
• C. $\pm (1+3\mathrm{i})$
• D. $\pm (3-\mathrm{i})$

Answer 1

The correct option is B

$\pm (1-3\mathrm{i})$

Explanation for the correct option:

Finding square root of given complex number

Given complex number : $\sqrt{-8-6\mathrm{i}}$

Let us assume this equal $\left(\mathrm{x}+\mathrm{i}\mathrm{y}\right)$,

$\Rightarrow \sqrt{-8-6\mathrm{i}}=\left(\mathrm{x}+\mathrm{iy}\right)$,

Squaring on both sides

$$\begin{gathered} \Rightarrow -8-6\mathrm{i}={(\mathrm{x}+\mathrm{iy})}^2 \\ \Rightarrow -8-6\mathrm{i}={\mathrm{x}}^2+i^2{\mathrm{y}}^2+2\mathrm{ixy} \\ \Rightarrow -8-6\mathrm{i}={\mathrm{x}}^2-{\mathrm{y}}^2+2\mathrm{ixy}\left[\because i^2=-1\right] \end{gathered}$$

Comparing the real and imaginary terms we get,

$$\begin{gathered} {\mathrm{x}}^2-{\mathrm{y}}^2=-8.....\left(i\right) \\ 2\mathrm{xy}=-6 \\ \Rightarrow y=-\frac{3}{x}...\left(ii\right) \end{gathered}$$

Substituting the value of $\mathrm{y}$ in equation $\left(i\right)$ to find the value of $\mathrm{x}$:

$\Rightarrow$ ${\mathrm{x}}^2-{\mathrm{y}}^2=-8$

$\Rightarrow$ ${\mathrm{x}}^2-\frac{9}{{\mathrm{x}}^2}=-8$

$\Rightarrow$ ${\mathrm{x}}^4-9=-8{\mathrm{x}}^2$

$$\begin{gathered} \Rightarrow {\mathrm{x}}^4+8{\mathrm{x}}^2-9=0 \\ \Rightarrow {\mathrm{x}}^4+9{\mathrm{x}}^2-{\mathrm{x}}^2-9=0 \end{gathered}$$

$$\begin{gathered} \Rightarrow {\mathrm{x}}^2\left({\mathrm{x}}^2+9\right)-1\left({\mathrm{x}}^2+9\right)=0 \\ \Rightarrow \left({\mathrm{x}}^2-1\right)\left({\mathrm{x}}^2+9\right)=0 \end{gathered}$$

Now,

$$\begin{gathered} \Rightarrow {\mathrm{x}}^2+9=0 \\ \Rightarrow \mathrm{x}=\sqrt{-9} \end{gathered}$$
$\Rightarrow \mathrm{x}=\pm 3i$ is an imaginary so it is not acceptable as $x$ is a real number.

Therefore, ${\mathrm{x}}^2-1=0\Rightarrow x=1,-1$

Substituting $x=1,-1$ in equation $\left(ii\right)$ we have

$\Rightarrow y=-3,3$

While substituting $\mathrm{x}$ and $\mathrm{y}$ value in this equation we get,

$\Rightarrow \sqrt{-8-6\mathrm{i}}=\pm \left(\mathrm{x}+\mathrm{iy}\right)$
$\Rightarrow \sqrt{-8-6\mathrm{i}}=\pm \left(1-3i\right)$

Hence, the correct answer is option (B).