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Roots of complex numbersEdexcel International A Level Further Maths: Flashcards

What these 12 flashcards ask

  • How many solutions has z^n=w (w\ne0)?
  • General argument of w=re^{i\theta}?
  • Formula for the nth roots of re^{i\theta}?
  • What do all the nth roots have in common?
  • Angle between neighbouring nth roots?
  • Roots of z^3=8?
  • Roots of z^3=1 in a+ib form?
  • Roots of z^n=1?
  • Sum of the nth roots of unity (n\ge2)?
  • Roots of z^3=-27i in the form re^{i\theta}?
  • Argument of -8+8\sqrt3\,i?
  • How do you solve (w+2)^3=-27i?

Exam questions on Roots of complex numbers

  1. Consider the equation z3=8z^3=8.
    Find the non-real roots of the equation in the form a+iba+ib, giving aa and bb as exact values.2 marks
  2. The complex number w=−8+83 iw=-8+8\sqrt3\,i and the equation z4=wz^4=w.
    Find the four roots of z4=wz^4=w in the form reiθre^{i\theta}, where −π<θ≤π-\pi<\theta\le\pi.2 marks
  3. Consider the equation z3=−27iz^3=-27i.
    Solve the equation, giving the roots in the form reiθre^{i\theta}, where −π<θ≤π-\pi<\theta\le\pi.3 marks
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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).