Calculus: Write z, z3, z7 and z−2 in the form a + b i.

Question

Let z = 7 e^2𝜋 i/3.

Write z, z³, z⁷ and z⁻² in the form a + b i

Answer 1

z^n = (r^n)(e^i*n*θ) = (r^n)(cosθn + i*sinθn) is the formula to use for all all them

z = z^1 = 7^1 (cos 2π/3 + i*sin 2π/3) = 7 (-1/2 + i*√3 /2) = -7/2 + 7* (sqrt(3)/2)*i

z^3 = (7^3) (cos 3*2π/3 + i*sin 3*2π/3) = (7^3)*(cos 2π + i*sin 2π) = 343*(1+ i*0) = 343

z^7 = (7^7) (cos 7*2π/3 + i*sin 7*2π/3) = (7^7) (cos 14π/3 + i*sin 14π/3) = (7^7) (cos 2π/3 + i*sin 2π/3)

You can then look up cos 2π/3 and sin 2π/3 on a unit circle, then plug them in the equation

z^-2 = (7^-2) (cos -2*2π/3 + i*sin -2*2π/3) = (1/49)*(cos -4π/3 + i*sin -4π/3) = (1/49)*(cos 2π/3 + i*sin 2π/3)

Plug the values of cos 2π/3 and sin 2π/3.

Cheers.