Math Is Fun Forum

Let x = z[sup]6[/sup].

Then x² - 56x - 512 = 0
∴ x = 64 or -8, by the quadratic equation.
∴ z[sup]6[/sup] = 64 or -8.

Consider the case when z[sup]6[/sup] = 64.

sin(kπ/3) is positive for k = 1 or 2, so in this case, z ∈ {1+i√3, i√3-1}.

In the case where z[sup]6[/sup] = -8, we can use similar reasoning to show that:

:D

This time, the appropriate k-values are  4 and 5, and so we get that z ∈ {[√(2)+i√(6)]/4, [-√(2)+i√(6)]/4}.

Those four values of z make up the complete solution.