How do I find the square roots of $i$ $i$ ?

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Answer 1 · 2014-10-26T09:59:54

Let $z = r e^{i θ}$ $z=re^{i theta}$ be the square-roots of $i$ $i$ .

$z^{2} = i \Rightarrow r^{2} e^{i (2 θ)} = e^{i (\frac{π}{2} + 2 n π)}$ $z^2=i Rightarrow r^2e^{i(2theta)}=e^{i(pi/2+2npi)}$

$Rightarrow {(r^2=1 Rightarrow r=1),(2theta=pi/2+2npi Rightarrow theta=pi/4+npi):}$

$z={e^{i pi/4}, e^{i {5pi}/4}}$

$={cos(pi/4)+isin(pi/4), cos({5pi}/4)+isin({5pi}/4)}$

$={1/sqrt{2}+1/sqrt{2}i, -1/sqrt{2}-1/sqrt{2}i}$

I hope that this was helpful.

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