Question

Quais são as raízes quartas de Z=16?

Answer 1

Z = 16 + 0 i
Wk = ρ^1/n[cos(θ+2kπ)/n + isen(θ + 2kπ)/n] , k ∈ Z

ρ = √(16² + 0²) => ρ = 16 
cosθ = a/ρ => cosθ = 16/16 = 1
senθ = b/ρ => senθ = 0/16 = 0    
cosθ =1 e senθ = 0 => θ = 0
p/ k = 0 => W0 = 16^1/4[cos(0 + 2.0.π)/4 + isen(0 + 2.0π)/4]
                  W0 = 2(cos0 + isen0)
                  w0 = 2(1 + 0) 
                  W0 = 2


p/ k = 1 => W1 = 16^1/4[cos(0 +2.1.π)/4 + isen(0 +2.1.π)/4]
                  W1 = 2(cosπ/2 + isenπ/2)
                  W1 = 2(0 + i.1)
                  W1 = 2i

p/ k = 2 => W2 = 16^1/4[cos(0 + 2.2.π)/4 + isen(0 + 2.2.π)/4]
                  W2 = 2(cosπ + isenπ)
                  W2 = 2(-1 + i.0)
                  W2 = -2

p/ k = 3 => W3 = 16^1/4[cos(0 + 2.3.π)/4 + isen(0 + 2.3.π)/4]
                  W3 = 2( cos3π/2 + isen3π/2)
                  Z4 = 2[(0 + i(-1)]
                  W3 = 2(-i)
                  W3 = -2i

S = { -2, 2, -2i, 2i }