### Instruction: Simplify $\left(\dfrac{-1+i\sqrt{3}}{2}\right)^6+\left(\dfrac{-1-i\sqrt{3}}{2}\right)^6.$.### Response: I notice that the two terms in the parentheses are complex numbers that are also roots of unity.
That means they are solutions to the equation $x^n=1$ for some positive integer $n$.
In fact, I recognize them as the third roots of unity, since they satisfy $x^3=1$.
Therefore, they are also sixth roots of unity, since $(x^3)^2=x^6=1$.
This means that raising them to the sixth power will give me 1 for both terms.
So the expression simplifies to $1+1=2$.