MHB Square Root Solutions for Complex Numbers

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The discussion focuses on finding square roots of complex numbers, specifically addressing part (ii) of a problem. A participant mentions obtaining the result 3$e^{\frac{1}{6}\theta i}$, prompting clarification on whether it should be 3$e^{\frac{1}{6}\pi i}$. The correct square roots of a complex number in polar form are identified as $\sqrt re^{\frac12\theta i}$ and $\sqrt re^{(\frac12\theta + \pi) i}$. This highlights the importance of accurately applying the polar representation of complex numbers in calculations. The conversation emphasizes the nuances in determining square roots of complex expressions.
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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 $\sqrt re^{(\frac12\theta+ \pi) i}$.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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