A) \[\pm (4+3\sqrt{-1})\]
B) \[\pm (3+4\sqrt{-1})\]
C) \[\pm (3-4\sqrt{-1})\]
D) \[\pm (4-3\sqrt{-1})\]
Correct Answer: C
Solution :
Now, \[\sqrt{-7-24\sqrt{-1}}\] \[=\sqrt{-1}\,\sqrt{7+24i}\] We know, \[\sqrt{a+ib}=\pm \left[ \sqrt{\frac{1}{2}[\sqrt{{{a}^{2}}+{{b}^{2}}}+a]} \right.\] \[+i\left. \sqrt{\frac{1}{2}\sqrt{{{a}^{2}}+{{b}^{2}}}-a} \right]\] \[\therefore \] \[i\sqrt{7+24\,i}=i\left[ \,\pm \left( \sqrt{\frac{1}{2}(\sqrt{49+576}+7)}+ \right. \right.\] \[\left. \left. i\sqrt{\frac{1}{2}(\sqrt{49+576}-7)} \right) \right]\] \[=i\left[ \pm \left( \sqrt{\frac{1}{2}(32)}+i\sqrt{\frac{1}{2}(18)} \right) \right]\] \[=i[\pm (4+3i)]=\pm (3-4\sqrt{-1})\]
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