Question
Solve the equation \(z^n = w\) for \(z\). Express the general solution in terms of \(|w|\) and \(\arg(w)\).
Hint
Hint
Recall that in polar form, \(z = r(\cos \theta + i\sin \theta)\) and apply De Moivre’s Theorem.
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Answer
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Explanation
- Formula Booklet
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Answer
Complete Answer
The general solution is \( z = \sqrt[n]{|w|} \left[ \cos\left(\frac{\arg(w)+2\pi k}{n}\right) + i\sin\left(\frac{\arg(w)+2\pi k}{n}\right) \right]\) for \(k = 0, 1, \ldots, n-1\).
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Answer Explanation