text: # Find the fourth roots of -17i?
  
  May 23, 2016
  
  Fourth roots of $- 17 i$ are $\sqrt[4]{17} {e}^{\frac{3 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{7 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{11 \pi}{8} i}$ and $\sqrt[4]{17} {e}^{\frac{15 \pi}{8} i}$
  
  #### Explanation:
  
  Let us wrote $- 17 i$ in polar form and as its modulus is $17$ and number is $117 \times \left(- i\right)$, we can write it as
  
  $- 17 i = 17 {e}^{\frac{3 \pi}{2} i}$
  
  Let $x = \sqrt[4]{- 17 i}$, then
  
  ${x}^{4} = - 17 i = 17 {e}^{\frac{3 \pi}{2} i}$
  
  Hence $x = \sqrt[4]{17} {e}^{\frac{3 \pi}{8} i}$
  
  Other roots can be obtained by adding $\frac{\pi}{2}$ to $\frac{3 \pi}{8}$ in succession (as they are one-fourth of $2 \pi$).
  
  i.e. $\sqrt[4]{17} {e}^{\frac{7 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{11 \pi}{8} i}$ and $\sqrt[4]{17} {e}^{\frac{15 \pi}{8} i}$
  
  To find them in the form of $a + b i$, you can use the following
  
  $r {e}^{i \theta} = r \cos \theta + i \sin \theta$
explanation: The document contains a math question and its corresponding answer. The question asks for the fourth roots of \(-17i\), and the answer provides the roots in exponential form and explains the process of finding them.
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distilabel_metadata.role: system
distilabel_metadata.content: # Find the fourth roots of -17i?
  
  May 23, 2016
  
  Fourth roots of $- 17 i$ are $\sqrt[4]{17} {e}^{\frac{3 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{7 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{11 \pi}{8} i}$ and $\sqrt[4]{17} {e}^{\frac{15 \pi}{8} i}$
  
  #### Explanation:
  
  Let us wrote $- 17 i$ in polar form and as its modulus is $17$ and number is $117 \times \left(- i\right)$, we can write it as
  
  $- 17 i = 17 {e}^{\frac{3 \pi}{2} i}$
  
  Let $x = \sqrt[4]{- 17 i}$, then
  
  ${x}^{4} = - 17 i = 17 {e}^{\frac{3 \pi}{2} i}$
  
  Hence $x = \sqrt[4]{17} {e}^{\frac{3 \pi}{8} i}$
  
  Other roots can be obtained by adding $\frac{\pi}{2}$ to $\frac{3 \pi}{8}$ in succession (as they are one-fourth of $2 \pi$).
  
  i.e. $\sqrt[4]{17} {e}^{\frac{7 \pi}{8} i}$, $\sqrt[4]{17} {e}^{\frac{11 \pi}{8} i}$ and $\sqrt[4]{17} {e}^{\frac{15 \pi}{8} i}$
  
  To find them in the form of $a + b i$, you can use the following
  
  $r {e}^{i \theta} = r \cos \theta + i \sin \theta$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding answer. The question asks for the fourth roots of \(-17i\), and the answer provides the roots in exponential form and explains the process of finding them.
  
  Label: question-and-answer