Expand, where i $\AA ^ { 2 } = - 1 .$
-$$( - 1 + i \sqrt { 3 } ) ^ { 3 }$$
A) 8
B) $- 1 - 3 \mathrm { i }$
C) $4 + 4 \mathrm { i }$
D) 1

Question

Quizplus · Accepted Answer

The given expression is $(-1 + i\sqrt{3})^3$. Recognize that $-1 + i\sqrt{3}$ is a complex form of the cube roots of unity other than 1. The cube roots of unity are $1, -\frac{1}{2} + i\frac{\sqrt{3}}{2}, -\frac{1}{2} - i\frac{\sqrt{3}}{2}$. Since $(-1 + i\sqrt{3})$ is one of the cube roots of unity, cubing it will give $1^3 = 1$. However, the correct calculation involves directly cubing the given expression, which actually results in $8$, not $1$. My explanation mistakenly applied the concept of roots of unity. Correctly cubing the expression involves algebraic expansion, leading to $8$.

Expand, Where I A˚2=−1.\AA ^ { 2 } = - 1 .A˚2=−1.

Expand, Where I $\AA ^ { 2 } = - 1 .$