Question

${\left(\frac{-1+i\sqrt{3}}{2}\right)}^{12}-{\left(\frac{-1-i\sqrt{3}}{2}\right)}^{12}=$

• A. 1
• B. 0
• C. 3
• D. 4

Answer 1

Answer: (B)

0

We know that
$w=\frac{-1+i\sqrt{3}i}{2}$ and $w^2=\frac{-1+i\sqrt{3}}{2}$ where $w,w^2$ are complex cube roots of unity.
Substituting in the above expression gives us
$(w^2{)}^{12}-w^{12}$
$=(w^3{)}^8-(w^3{)}^4$
$=1-1$
$=0$... since $w^3=1$