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Powers of Complex Numbers (DeMoivre's Theorem) quiz

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  • What does De Moivre's Theorem state for raising a complex number in polar form to a power n?
    Raise the modulus r to the nth power and multiply the angle θ by n: (r cis θ)^n = r^n cis(nθ).
  • How do you multiply two complex numbers in polar form?
    Multiply their r values and add their angles.
  • What is the shortcut provided by De Moivre's Theorem when raising a complex number to a power?
    Raise the modulus to the power and multiply the angle by the power.
  • If z = 3 cis 15°, what is z^2 using De Moivre's Theorem?
    z^2 = 9 cis 30°.
  • Why is De Moivre's Theorem useful for higher powers of complex numbers?
    It simplifies calculations that would otherwise require repeated multiplication.
  • How do you find the nth root of a complex number in polar form?
    Take the nth root of r and divide the angle by n, then add 360°k/n (or 2πk/n) for all integer k from 0 to n-1.
  • What formula do you use to find all nth roots of a complex number?
    z_k = r^(1/n) cis[(θ + 360°k)/n] for k = 0, 1, ..., n-1.
  • How many distinct nth roots does a complex number have?
    It has n distinct roots.
  • What does the variable k represent when finding roots of complex numbers?
    k is an integer from 0 to n-1, used to find all possible roots.
  • If you want the cube roots of 8 cis 45°, what is the modulus of each root?
    The modulus is 2, since 8^(1/3) = 2.
  • How do you calculate the angles for the cube roots of 8 cis 45°?
    Use θ_k = (45° + 360°k)/3 for k = 0, 1, 2.
  • What are the three angles for the cube roots of 8 cis 45°?
    15°, 135°, and 255°.
  • What is the general form for the kth root of a complex number in polar form?
    z_k = r^(1/n) cis[(θ + 360°k)/n].
  • Why do you add 360°k (or 2πk) when finding roots of complex numbers?
    To account for all possible solutions around the unit circle.
  • What is the abbreviation 'cis' short for in complex numbers?
    'cis θ' stands for 'cos θ + i sin θ'.