Evaluate (5+5√3 i)^7 using DeMoivre’s theorem? write your answer in rectangular form

Answer 1

De Moivre's formula says:

(cos x + i sin x)ⁿ = cos(nx) + i sin(nx)

We need to convert 5 + 5sqrt(3)i to polar form. We have:

x = arctan(5sqrt(3)/5) = arctan(sqrt(3)) = π/3

r = sqrt(25(3) + 25) = sqrt(100) = 10

Therefore, we have:

(10(cos(π/3) + i sin(π/3)))⁷ = 10⁷(cos(π/3) + i sin(π/3))⁷ = 10⁷(cos(7π/3) + i sin(7π/3)) = 10⁷(1 + i sqrt(3))/2