Animated Solution for Mathematics - Complex Numbers: If $\omega (\ne 1)$ be a cube root of unity and $(1+{\omega }^2{)}^n=(1+{\omega }^4{)}^n$, then the least positive value of $n$ is

The cube roots of unity, denoted as $1$, $\omega$, and ${\omega }^2$, are geometric anchors on the complex plane. They are perfectly spaced at $120^{\circ }$ intervals on the unit circle.

Our primary objective is to solve the equation:

The most powerful tool in our arsenal is the fundamental identity:

This identity arises because the sum of the roots of the polynomial $x^3-1=0$ is zero, as there is no $x^2$ term in the expansion.

First, we simplify the Left Hand Side (LHS). Using the identity $1+{\omega }^2=-\omega$, the expression becomes:

Next, we simplify the Right Hand Side (RHS). Since ${\omega }^3=1$, we know that ${\omega }^4={\omega }^3\cdot \omega =\omega$. Applying the identity $1+\omega =-{\omega }^2$, the expression becomes:

Equating the simplified sides, we have:

Dividing both sides by $(-\omega {)}^n$, we obtain:

Using the laws of exponents, this simplifies to:

We are looking for the least positive integer $n$ such that ${\omega }^n=1$. Given the property ${\omega }^3=1$, the smallest positive integer that satisfies this condition is: