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

Options

A. 2
B. 3
C. 5
D. 6

Explanation

To solve the equation \((1 + \omega^2)^n = (1 + \omega^4)^n\) where \(\omega\) is a cube root of unity, we can follow these steps:

### Step 1: Understand the properties of cube roots of unity
The cube roots of unity are given by:
1. \(\omega^3 = 1\)
2. \(1 + \omega + \omega^2 = 0\)

From the second property, we can express \(\omega^2\) as:
\[ \omega^2 = -1 - \omega \]

### Step 2: Rewrite the equation
We start from the equation:
\[ (1 + \omega^2)^n = (1 + \omega^4)^n \]
Since \(\omega^4 = \omega\) (because \(\omega^4 = \omega^{3+1} = \omega\)), we can rewrite the equation as:
\[ (1 + \omega^2)^n = (1 + \omega)^n \]

### Step 3: Simplify the expression
We can divide both sides by \((1 + \omega)^n\) (assuming \(1 + \omega \neq 0\)):
\[ \frac{1 + \omega^2}{1 + \omega} = 1 \]
This implies:
\[ 1 + \omega^2 = 1 + \omega \]

### Step 4: Solve for \(\omega\)
Subtracting 1 from both sides gives:
\[ \omega^2 = \omega \]
This is only true if \(\omega = 0\) or if \(\omega\) is a specific value. However, since \(\omega\) is a cube root of unity, we can check the values of \(\omega\):
- If \(\omega = 1\), then \(\omega^2 = 1\) (not valid since \(\omega \neq 1\)).
- If \(\omega = \omega\), then \(\omega^2 = \omega^2\) (valid).

### Step 5: Analyze the implications
Since \(1 + \omega^2\) and \(1 + \omega\) are equal, we can conclude that the equality holds for certain values of \(n\).

### Step 6: Find the least positive value of \(n\)
The equality \((1 + \omega^2)^n = (1 + \omega)^n\) holds when:
\[ \left(\frac{1 + \omega^2}{1 + \omega}\right)^n = 1 \]
This means that the ratio must be a root of unity. The least positive value of \(n\) that satisfies this condition is when \(n\) is a multiple of 3, since the cube roots of unity repeat every 3.

Thus, the least positive value of \(n\) is:
\[ \boxed{3} \]