Question

Please help me do this question

Answer 1

If
`z^4 = -3 + 4j`, where
`j = √(-1)`, we first write
`z^4` in exponential form using

`\left|z^4\right| = √((-3)^2 + 4^2) = √(25) = 5`

and

`\arg\left(z^4\right) = \pi + \tan^(-1)\left(-\frac43\right) = \pi - \tan^(-1)\left(\frac43\right)`

Then

`z^4 = 5 \exp\left(j \left(\pi - \tan^(-1)\left(\frac43\right)\right)`

By de Moivre's theorem, the fourth roots of
`z^4` are

`z = 5^(1/4) \exp\left(j \frac{\pi - \tan^(-1)\left(\frac43\right) + 2k\pi}4\right)`

with
`k\in\{0,1,2,3\}`, so the four possible values of
`z` are

`k = 0 \implies z_1 = 5^(1/4) \exp\left(j \frac{\pi - \tan^(-1)\left(\frac43\right)}4\right)`

`k = 1 \implies z_2 = 5^(1/4) \exp\left(j \frac{3\pi - \tan^(-1)\left(\frac43\right)}4\right)`

`k = 2 \implies z_3 = 5^(1/4) \exp\left(j \frac{5\pi - \tan^(-1)\left(\frac43\right)}4\right)`

`k = 3 \implies z_4 = 5^(1/4) \exp\left(j \frac{7\pi - \tan^(-1)\left(\frac43\right)}4\right)`

(see attached plot from WolframAlpha)