Direct nth roots: general complex RHS

1 Find the roots of the equation \(z ^ { 3 } = 7 \sqrt { 3 } - 7 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi \leqslant \theta < \pi\).

1 Find the roots of the equation \(z ^ { 3 } = - 108 \sqrt { 3 } + 108\) i, giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).

3. Solve the equation $$z ^ { 5 } = 16 - 16 \mathrm { i } \sqrt { 3 }$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(\theta\) is in terms of \(\pi\) and \(0 \leqslant \theta < 2 \pi\).

4. (a) Express the complex number \(18 \sqrt { 3 } - 18 \mathrm { i }\) in the form $$r ( \cos \theta + \mathrm { i } \sin \theta ) \quad - \pi < \theta \leqslant \pi$$ (b) Solve the equation $$z ^ { 4 } = 18 \sqrt { 3 } - 18 i$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(- \pi < \theta \leqslant \pi\)

3. (a) Express the complex number \(- 2 + ( 2 \sqrt { 3 } ) \mathrm { i }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).
(b) Solve the equation $$z ^ { 4 } = - 2 + ( 2 \sqrt { } 3 ) i$$ giving the roots in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).

6. Solve the equation $$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta < \pi\).

3. (a) Find the four roots of the equation \(z ^ { 4 } = 8 ( \sqrt { 3 } + \mathrm { i } )\) in the form \(z = r \mathrm { e } ^ { \mathrm { i } \theta }\) (b) Show these roots on an Argand diagram.

3. Solve the equation $$z ^ { 3 } + 32 + 32 i \sqrt { 3 } = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)

Solve the equation $$z ^ { 3 } = 4 \sqrt { } 2 - 4 \sqrt { } 2 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(- \pi < \theta \leqslant \pi\).

2

1. Solve the equation \(z ^ { 4 } = 2 ( 1 + \mathrm { i } \sqrt { 3 } )\), giving the roots exactly in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Sketch an Argand diagram to show the lines from the origin to the point representing \(2 ( 1 + i \sqrt { 3 } )\) and from the origin to the points which represent the roots of the equation in part (i).

Solve the equation $$z^2 = 4\sqrt{2} - 4\sqrt{2}i,$$ giving your answers in the form \(r(\cos \theta + i \sin \theta)\), where \(-\pi < \theta \leq \pi\). [6]

\(z = -8 + (8\sqrt{3})i\)

1. Find the modulus of \(z\) and the argument of \(z\). [3]

Using de Moivre's theorem,

1. find \(z^3\). [2]
2. find the values of \(w\) such that \(w^4 = z\), giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [5]

Find the cube roots of the complex number \(z = 11 - 2i\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and correct to three decimal places. [7]

Find the three cube roots of the complex number \(2 + 3i\), giving your answers in Cartesian form. [9]