Direct nth roots: purely real or imaginary RHS

Solve z^n = w where w is a real number or purely imaginary (e.g. z^5 = 32, z^3 = -64i), requiring straightforward De Moivre application with simple modulus and argument.

5 questions · Moderate -0.2

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Edexcel F2 2017 June Q1
5 marks Moderate -0.8
  1. Solve the equation
$$z ^ { 5 } = 32$$ Give your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\)
Edexcel F2 2021 October Q1
4 marks Standard +0.3
  1. Solve the equation
$$z ^ { 5 } - 32 i = 0$$ giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < 2 \pi\)
OCR Further Pure Core 1 2020 November Q4
4 marks Moderate -0.3
4 In this question you must show detailed reasoning.
  1. Determine the square roots of 25 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
  2. Illustrate the number 25i and its square roots on an Argand diagram.
AQA FP2 2014 June Q1
7 marks Standard +0.3
1
  1. Express - 9 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [2 marks]
  2. Solve the equation \(z ^ { 4 } + 9 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [5 marks]
OCR MEI Further Pure Core 2023 June Q5
7 marks Moderate -0.3
5
  1. In this question you must show detailed reasoning.
    Determine the sixth roots of - 64 , expressed in \(r \mathrm { e } ^ { \mathrm { i } \theta }\) form.
  2. Represent the roots on an Argand diagram.