\(\sqrt{i}\)


How many square roots of \(i\) are there ? 

To check out this question, we may move on and use the ln manipulation like follows: 

\(\sqrt{i} = i^{\frac{1}{2}} \)

\(\Rightarrow i^{\frac{1}{2}} =  e^{ln\left ( i^{\frac{1}{2}} \right )} = e^{\frac{1}{2}ln\left ( i \right )} \)

But, we knew that \( ln\left ( i \right ) = i\frac{\pi }{2}\) (Click here for more ...)    

Thus, \( e^{\frac{1}{2}ln\left ( i \right )} = e^{i\frac{\pi }{4}}\) 


Now, since the root \(e^{i\frac{\pi }{4}}\) we found have the exponential form, then the Euler's Formula can be used also to find the equivalent roots. 

Hence, \( \sqrt{i} = e^{i\frac{\pi }{4}} = \frac{1}{\sqrt{2}} +i\frac{1}{\sqrt{2}} \)