Square Root Solutions for Complex Numbers

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SUMMARY

The discussion centers on finding square roots of complex numbers, specifically using the polar form representation. The square roots of a complex number expressed as \( re^{\theta i} \) are definitively given as \( \sqrt{r}e^{\frac{1}{2}\theta i} \) and \( \sqrt{r}e^{(\frac{1}{2}\theta + \pi)i} \). A participant noted a specific case where they calculated one of the square roots as \( 3e^{\frac{1}{6}\theta i} \), prompting clarification on the correct angle representation.

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Milly
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View attachment 4273Helppp for part (ii). I got 3$e^{\frac{1}{6}\theta i}$
 

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Milly said:
Helppp for part (ii). I got 3$e^{\frac{1}{6}\theta i}$ Do you mean $\color{red}{3e^{\frac{1}{6}\pi i}}$?
The two square roots of $re^{\theta i}$ are $\sqrt re^{\frac12\theta i}$ and $\sqrt re^{(\frac12\theta+ \pi) i}$.
 

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