Explanation:
We can simplify this problem using De Moivre's theorem. De Moivre's theorem states that:
(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
Here, z = 1 + √3i, so we can express z in polar form as:
z = 2(cos π/3 + i sin π/3)
To find z^6, we can apply De Moivre's theorem:
z^6 = [2(cos π/3 + i sin π/3)]^6
= 2^6(cos 6π/3 + i sin 6π/3)
= 2^6(cos 2π + i sin 2π)
= 2^6(1 + 0i)
= 64
Therefore, z^6 = 64.