Write each expression in the standard form a+bi

Question

(3 + 4i)^4? using De Moivre's theorem.

 

Answer 1

The expression is (3+4i)⁴.

Write the complex number z = x + iy in polar from z = r[cos(θ) + i sin(θ)].

r = √(x² + y²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

Here, x = 3 >0, so θ = tan^-1(y/x) = tan^-1(4/3) = tan^-1(4/3) = 53.13^o.

Therefore 3 + 4i = 5[cos(53.13^o) + i sin(53.13^o)].

(3 + 4i)⁴ = {5[cos(53.13^o) + i sin(53.13^o)]}⁴

Apply De Moivre's theorem: If z = r[cos(θ) + i sin(θ)] ⇒ zⁿ = rⁿ[cos(nθ) + i sin(nθ)].

(3 + 4i)⁴ = 5⁴[cos(4*53.13^o) + i sin(4*53.13^o)]

(3 + 4i)⁴ = 625[cos(212.52^o) + i sin(212.52^o)]

(3 + 4i)⁴ = 625[-0.8432 + i (-0.5376)]

(3 + 4i)⁴ = - 527 - i 336.