Question

The square roots of $-7-24\sqrt{-1}$ are

• A. $\pm (4+3\sqrt{-1})$
• B. $\pm (3+4\sqrt{-1})$
• C. $\pm (3-4\sqrt{-1})$
• D. $\pm (4-3\sqrt{-1})$

Answer 1

Answer: (C)

Now, $\sqrt{-7-24\sqrt{-1}}$
$=\sqrt{-1}\sqrt{7+24i}$
We know, $\sqrt{a+ib}=\pm [\sqrt{\frac{1}{2}[\sqrt{a^2+b^2}+a]}$
$+i\sqrt{\frac{1}{2}\sqrt{a^2+b^2}-a}]$
$\therefore$ $i\sqrt{7+24i}=i[\pm (\sqrt{\frac{1}{2}(\sqrt{49+576}+7)}+$
$i\sqrt{\frac{1}{2}(\sqrt{49+576}-7)})]$
$=i\left[\pm \left(\sqrt{\frac{1}{2}(32)}+i\sqrt{\frac{1}{2}(18)}\right)\right]$
$=i[\pm (4+3i)]=\pm (3-4\sqrt{-1})$