Complex Numbers
Cube Roots of Unity — Finding m
nta_pyq_2026_jan
Grade 11
Question:
Let $\alpha=\dfrac{-1+i\sqrt{3}}{2}$ and $\beta=\dfrac{-1-i\sqrt{3}}{2}$, $i=\sqrt{-1}$. If $(7-7\alpha+9\beta)^{20}+(9+7\alpha-7\beta)^{20}+(-7+9\alpha+7\beta)^{20}+(14+7\alpha+7\beta)^{20}=m^{10}$, then $m$ is _____
Step-by-Step Solution
Key Concept: $\alpha=\omega,\beta=\omega^2$ (cube roots of unity). Compute $z_1=7-7\omega+9\omega^2$, $z_2=9+7\omega-7\omega^2$, $z_3=-7+9\omega+7\omega^2$. Note $z_2=z_1\omega$, $z_3=z_1\omega^2$.
First three terms sum to 0. Fourth term $=7^{20}=49^{10}$. $m=49$.
Correct Answer: 49