Determine and examine the 𝑛th roots of complex numbers and their location in the complex plane.

Consider the complex number \( z = 8\operatorname{cis}\left(\frac{2\pi}{3}\right) \). (a) Determine the three cube roots of \( z \), expressing each in the form \( r\operatorname{cis}(\theta) \) where \( 0 \le \theta < 2\pi \). (2 marks) (b) Represent these three roots on an Argand diagram. (1 mark)

Consider the complex number \( z = 8\text{cis}\left(\frac{2\pi}{3}\right) \). (a) Determine all three cube roots of \( z \). Express your answers in polar form \( r\text{cis}(\theta) \), where \( 0 \leq \theta < 2\pi \). (3 marks) (b) Using the table provided, represent each cube root of \( z \) on an Argand diagram and describe the geometric relationship between these roots. (2 marks)

Consider the complex number \(z = 8\operatorname{cis}\left(\frac{2\pi}{3}\right)\). Determine the modulus of each cube root of \(z\). (1 mark) Hence determine the three cube roots of \(z\), expressing your answers in the form \(r\operatorname{cis}(\theta)\). (1 mark)

The diagram shows the complex number \(z = 8\operatorname{cis}\left(\frac{2\pi}{3}\right)\) plotted in the complex plane. (a) Determine all cube roots of \(z\), expressing your answers in the form \(r\operatorname{cis}(\theta)\) where \(0 \leq \theta < 2\pi\). (2 marks) (b) Verify that these roots form the vertices of an equilateral triangle in the complex plane. (1 mark)

The table below shows three complex numbers and their corresponding values of \(n\) for finding \(n\)th roots. **Part (a)** (2 marks) Determine all fourth roots of \(z_1\), expressing your answers in the form \(r\operatorname{cis}(\theta)\). **Part (b)** (3 marks) Determine all cube roots of \(z_3\), expressing your answers in Cartesian form \(x + yi\). Provide an Argand diagram showing the location of these roots as part of your solution.

Let \( z = 8\operatorname{cis}\left(\frac{2\pi}{3}\right) \). (a) Determine the three cube roots of \( z \), expressing your answers in the form \( r\operatorname{cis}(\theta) \) where \( 0 \leq \theta < 2\pi \). (2 marks) (b) Represent these three roots on an Argand diagram. (1 mark)

Consider the complex number \( z = 8\operatorname{cis}\left(\frac{2\pi}{3}\right) \). (a) Determine the three cube roots of \( z \), expressing your answers in the form \( r\operatorname{cis}(\theta) \) where \( -\pi < \theta \leq \pi \). (b) Represent these three cube roots on an Argand diagram.

The diagram shows a complex number z plotted in the complex plane. (a) Determine the three cube roots of z, expressing your answers in the form r cis(θ) where 0 ≤ θ < 2π. (3 marks) (b) Verify that the sum of the three cube roots equals zero. (1 mark)

Given \(z = 8\operatorname{cis}\left(\frac{2\pi}{3}\right)\), what is the principal argument of the principal cube root of \(z\)?