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 \left[ \sqrt{\frac{1}{2}[\sqrt{{{a}^{2}}+{{b}^{2}}}+a]} \right. $
$ +i\left. \sqrt{\frac{1}{2}\sqrt{{{a}^{2}}+{{b}^{2}}}-a} \right] $
$ \therefore $ $ i\sqrt{7+24\,i}=i\left[ \,\pm \left( \sqrt{\frac{1}{2}(\sqrt{49+576}+7)}+ \right. \right. $
$ \left. \left. i\sqrt{\frac{1}{2}(\sqrt{49+576}-7)} \right) \right] $
$ =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}) $