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"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.

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Let \(z=x+iy\) be a complex number where \(x\) and \(y\) are integers. Find the area of the rectangle whose vertices are the roots of the equation \[\large z \bar{z}^3+\bar{z}z^3=350.\]

**Details:** \( \bar{z}=x-iy\) and \(i=\sqrt{-1}\).

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How many complex numbers are the conjugate of their own cube?

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The complex numbers \(p\) and \(q\) satisfy \(p^3=5+i\sqrt{2}\) and \(q^3=5-i\sqrt{2}\).

Find the only possible integer value of \(p+q\).

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\[\large e ^{ i\theta }e^{ 2i\theta} e^{ 3i\theta }...\large e^{ ni\theta } = 1\]

Find the value of \(\theta\) satisfying the above equation for \(m\in\mathbb{N}\).

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