$$(1 + \sqrt{3i} ) z^{4} = - 800$$
I tried doing this, but it feels wrong:
Edit: I also tried multiplying with the conjugate in step 2, and then multiplying with the conjugate again to get 9 in the denominator. It didn't help.
What is the right approach for solving equations like these?
In your second step, try multiplying the right hand side by the complex conjugate of the denominator (divided by itself, so you're multiplying by $1$). You should be able to reduce it to a simplified $a + bi$ that you can take the 4th root of $z$ to solve.