text: 1. ## De moivres theorem.
  
  Use DMT to solve the following equations in polar form.
  
  $z^3=-1-i$
  $z^3=-4+4\sqrt{3i}$
  
  2. What have you tried so far?. These problems are just routine knowing the corresponding theory.
  
  3. I'll show you how to do one, then you try to do the other...
  
  $\displaystyle |-1-i| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}$.
  
  Note that this complex number is in the third quadrant. The angle in the first quadrant is $\displaystyle \arctan{\frac{1}{1}} = \frac{\pi}{4}$.
  So the angle in the third quadrant, $\displaystyle \arg{(-1 - i)} = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$.
  
  $\displaystyle z^3 = \sqrt{2}\,\textrm{cis}\,{\left(\frac{5\pi}{4}\righ t)}$
  
  $\displaystyle z = (\sqrt{2})^{\frac{1}{3}}\,\textrm{cis}\,{\left(\fr ac{1}{3}\cdot \frac{5\pi}{4}\right)}$
  
  $\displaystyle = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12}\right)}$.
  
  There are always three cube roots, and they are of the same magnitude and evenly spaced about a circle. So they differ by an angle of $\displaystyle \frac{2\pi}{3}$.
  
  So the other solutions are $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12}\right)}$
  
  and $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{21\pi}{12}\right)}$.
  
  But since $\displaystyle -\pi < \arg{z} \leq \pi$, we rewrite this as
  
  $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(-\frac{11\pi}{12}\right)}$ and $\displaystyle \sqrt[6]{2}\,{\left(-\frac{3\pi}{12}\right)}$.
explanation: The document contains a math problem involving De Moivre's Theorem and the solution to one of the given equations. It also includes a question asking what the user has tried so far and a detailed step-by-step solution for one of the equations.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 1. ## De moivres theorem.
  
  Use DMT to solve the following equations in polar form.
  
  $z^3=-1-i$
  $z^3=-4+4\sqrt{3i}$
  
  2. What have you tried so far?. These problems are just routine knowing the corresponding theory.
  
  3. I'll show you how to do one, then you try to do the other...
  
  $\displaystyle |-1-i| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}$.
  
  Note that this complex number is in the third quadrant. The angle in the first quadrant is $\displaystyle \arctan{\frac{1}{1}} = \frac{\pi}{4}$.
  So the angle in the third quadrant, $\displaystyle \arg{(-1 - i)} = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$.
  
  $\displaystyle z^3 = \sqrt{2}\,\textrm{cis}\,{\left(\frac{5\pi}{4}\righ t)}$
  
  $\displaystyle z = (\sqrt{2})^{\frac{1}{3}}\,\textrm{cis}\,{\left(\fr ac{1}{3}\cdot \frac{5\pi}{4}\right)}$
  
  $\displaystyle = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12}\right)}$.
  
  There are always three cube roots, and they are of the same magnitude and evenly spaced about a circle. So they differ by an angle of $\displaystyle \frac{2\pi}{3}$.
  
  So the other solutions are $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12}\right)}$
  
  and $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{21\pi}{12}\right)}$.
  
  But since $\displaystyle -\pi < \arg{z} \leq \pi$, we rewrite this as
  
  $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(-\frac{11\pi}{12}\right)}$ and $\displaystyle \sqrt[6]{2}\,{\left(-\frac{3\pi}{12}\right)}$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem involving De Moivre's Theorem and the solution to one of the given equations. It also includes a question asking what the user has tried so far and a detailed step-by-step solution for one of the equations.
  
  Label: multiple-question-and-answer