We are given $B = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ and $C = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$. We need to compute $(B^2 - C + 1)^{20}$.

1. Problem Description

We are given $B = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ and $C = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$. We need to compute $(B^2 - C + 1)^{20}$.

2. Solution Steps

First, we find $B^2$.

$B^2 = \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)^2 = \left(-\frac{1}{2}\right)^2 + 2\left(-\frac{1}{2}\right)\left(i\frac{\sqrt{3}}{2}\right) + \left(i\frac{\sqrt{3}}{2}\right)^2$

$B^2 = \frac{1}{4} - i\frac{\sqrt{3}}{2} - \frac{3}{4} = -\frac{2}{4} - i\frac{\sqrt{3}}{2} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$

Next, we compute $B^2 - C + 1$.

$B^2 - C + 1 = \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) - \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) + 1$

$B^2 - C + 1 = -\frac{1}{2} - i\frac{\sqrt{3}}{2} + \frac{1}{2} + i\frac{\sqrt{3}}{2} + 1 = 1$

Then, we compute $(B^2 - C + 1)^{20}$.

$(B^2 - C + 1)^{20} = (1)^{20} = 1$

3. Final Answer

The final answer is 1.