Answer:
<em>See below.</em>
Step-by-step explanation:
To find roots of an equation, we use this formula:
where k = 0, 1, 2, 3... (n = root; equal to n - 1; dependent on the amount of roots needed - 0 is included).
In this case, n = 4.
Therefore, we adjust the polar equation we are given and modify it to be solved for the roots.
Part 2: Solving for root #1
To solve for root #1, make k = 0 and substitute all values into the equation. On the second step, convert the measure in degrees to the measure in radians by multiplying the degrees measurement by
and simplify.
![z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(0)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(0)\pi}{4}))](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D16%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%28cos%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%280%29%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%280%29%5Cpi%7D%7B4%7D%29%29)
![z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{\pi}{4}))](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%29)
![z^{\frac{1}{4}} = 2(sin(\frac{5\pi}{18}+\frac{\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{\pi}{4}))](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%20%3D%202%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%29)
<u>Root #1:</u>
![\large\boxed{z^\frac{1}{4}=2(cos(\frac{19\pi}{36}))+\mathfrack{i}(sin(\frac{19\pi}{38}))}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7Bz%5E%5Cfrac%7B1%7D%7B4%7D%3D2%28cos%28%5Cfrac%7B19%5Cpi%7D%7B36%7D%29%29%2B%5Cmathfrack%7Bi%7D%28sin%28%5Cfrac%7B19%5Cpi%7D%7B38%7D%29%29%7D)
Part 3: Solving for root #2
To solve for root #2, follow the same simplifying steps above but change <em>k</em> to k = 1.
![z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(1)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(1)\pi}{4}))](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D16%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%28cos%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%281%29%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%281%29%5Cpi%7D%7B4%7D%29%29)
![z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{2\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{2\pi}{4}))\\](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B2%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B2%5Cpi%7D%7B4%7D%29%29%5C%5C)
![z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{\pi}{2}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{\pi}{2}))\\](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%29%29%5C%5C)
<u>Root #2:</u>
![\large\boxed{z^{\frac{1}{4}}=2(cos(\frac{7\pi}{9}))+\mathfrak{i}(sin(\frac{7\pi}{9}))}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7Bz%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B7%5Cpi%7D%7B9%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B7%5Cpi%7D%7B9%7D%29%29%7D)
Part 4: Solving for root #3
To solve for root #3, follow the same simplifying steps above but change <em>k</em> to k = 2.
![z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(2)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(2)\pi}{4}))](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D16%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%28cos%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%282%29%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%282%29%5Cpi%7D%7B4%7D%29%29)
![z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{4\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{4\pi}{4}))\\](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B4%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B4%5Cpi%7D%7B4%7D%29%29%5C%5C)
![z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\pi))+\mathfrak{i}(sin(\frac{5\pi}{18}+\pi))\\](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cpi%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cpi%29%29%5C%5C)
<u>Root #3</u>:
![\large\boxed{z^{\frac{1}{4}}=2(cos(\frac{23\pi}{18}))+\mathfrak{i}(sin(\frac{23\pi}{18}))}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7Bz%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B23%5Cpi%7D%7B18%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B23%5Cpi%7D%7B18%7D%29%29%7D)
Part 4: Solving for root #4
To solve for root #4, follow the same simplifying steps above but change <em>k</em> to k = 3.
![z^{\frac{1}{4}}=16^{\frac{1}{4}}(cos(\frac{200}{4}+\frac{2(3)\pi}{4}))+\mathfrak{i}(sin(\frac{200}{4}+\frac{2(3)\pi}{4}))](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D16%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%28cos%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%283%29%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B200%7D%7B4%7D%2B%5Cfrac%7B2%283%29%5Cpi%7D%7B4%7D%29%29)
![z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{6\pi}{4}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{6\pi}{4}))\\](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B6%5Cpi%7D%7B4%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B6%5Cpi%7D%7B4%7D%29%29%5C%5C)
![z^{\frac{1}{4}}=2(cos(\frac{5\pi}{18}+\frac{3\pi}{2}))+\mathfrak{i}(sin(\frac{5\pi}{18}+\frac{3\pi}{2}))\\](https://tex.z-dn.net/?f=z%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B3%5Cpi%7D%7B2%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B5%5Cpi%7D%7B18%7D%2B%5Cfrac%7B3%5Cpi%7D%7B2%7D%29%29%5C%5C)
<u>Root #4</u>:
![\large\boxed{z^{\frac{1}{4}}=2(cos(\frac{16\pi}{9}))+\mathfrak{i}(sin(\frac{16\pi}{19}))}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7Bz%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D2%28cos%28%5Cfrac%7B16%5Cpi%7D%7B9%7D%29%29%2B%5Cmathfrak%7Bi%7D%28sin%28%5Cfrac%7B16%5Cpi%7D%7B19%7D%29%29%7D)
The fourth roots of <em>16(cos 200° + i(sin 200°) </em>are listed above.