What is the modulus of $\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?$

If x² + x + 1 = 0, then what is the value of x¹⁹⁹ + x²⁰⁰ + x²⁰¹? 

What is $\rm \sum\limits_{n=1}^{8n+7} i^n$ equal to, where i = √-1?

If z = x + iy, where i = √-1, then what does the equations zz̅ + ∣z∣² + 4(z + z̅) - 48 = 0 represent?