First slide

Algebra of complex numbers

Question

If $ω (\neq 1)$ is a complex cube root of unity and ${(1 + ω^{4})}^{n} = {(1 + ω^{8})}^{n}$ then the least positive integral value of $n$ is

Moderate

A

2
B

3
C

6
D

12

Solution

As $ω^{4} = ω, ω^{8} = ω^{2}$ we get
$\begin{array}{l} (1 + ω)^{n} = {(1 + ω^{2})}^{n} \\ \Rightarrow {(- ω^{2})}^{n} = (- ω)^{n} \end{array}$
$\begin{array}{lr} \Rightarrow ω^{n} = 1 \\ ∴ n = 3 \end{array}$

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