Question

`w^4=-16i` how to write in de moivre theorem form?

Answer 1

See answer below

Explanation:

Given that

`w^4=-16i`

`w^4=16(-i)`

`w^4=16(\cos(-\pi/2)+i\sin(-\pi/2))`

`w=(16(\cos(-\pi/2)+i\sin(-\pi/2)))^{1/4}`

`=16^{1/4}(\cos(-\pi/2)+i\sin(-\pi/2))^{1/4}`

`=2(\cos(2k\pi-\pi/2)+i\sin(2k\pi-\pi/2))^{1/4}`

`=2(\cos(\frac{(4k-1)\pi}{2})+i\sin(\frac{(4k-1)\pi}{2}))^{1/4}`

`=2(\cos(\frac{(4k-1)\pi}{8})+i\sin(\frac{(4k-1)\pi}{8}))`

Where, `k=0, 1, 2, 3`

By setting values of `k`, we get four values/roots of given `4`th degree equation.