Direct nth roots: find and express roots

Questions that ask solely to solve z^n = w and express all n roots in a specified form (exponential, polar, or Cartesian), with no significant follow-up tasks beyond possibly plotting on an Argand diagram.

5 questions · Standard +0.4

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Edexcel FP2 2006 January Q5
5 marks Standard +0.3
5. Solve the equation \(z ^ { 5 } = \mathrm { i }\) giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\).
(Total 5 marks)
OCR FP3 2009 June Q1
4 marks Standard +0.3
1 Find the cube roots of \(\frac { 1 } { 2 } \sqrt { 3 } + \frac { 1 } { 2 } \mathrm { i }\), giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(0 \leqslant \theta < 2 \pi\).
AQA FP2 2008 January Q1
8 marks Standard +0.3
1
  1. Express \(4 + 4 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  2. Solve the equation $$z ^ { 5 } = 4 + 4 i$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
OCR FP3 Q1
4 marks Standard +0.8
Find the cube roots of \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), giving your answers in the form \(\cos \theta + i \sin \theta\), where \(0 \leqslant \theta < 2\pi\). [4]
AQA Further Paper 2 Specimen Q4
4 marks Standard +0.3
Solve the equation \(z^3 = i\), giving your answers in the form \(e^{i\theta}\), where \(-\pi < \theta \leq \pi\) [4 marks]