Solve the following equation: Z^4=3-4i

Question

Complex number

Answer 1

Let z=reꜞᶿ=r(cosθ+isinθ).

z⁴=r⁴e⁴ꜞᶿ=r⁴(cos4θ+isin4θ)=3-4i.

r⁴cos4θ=3

r⁴sin4θ=-4

Dividing we get: tan4θ=-4/3 from which sin4θ=-4/5, cos4θ=3/5.

Squaring and adding: r⁸=25, r⁴=5, r=∜5=1.4953 approx.

4θ=-0.9273 approx, θ=-0.2318; sinθ=-0.2298, cosθ=0.9732.

z=1.4953(0.9732-0.2298i)=1.4552-0.3436i.