Question

if w be an imaginary cube root of unity and (1+w^2)^n= (1+w^4)^n, then the least positive integral value of n is(w is not equal to 1)​

Answer 1

Answer:

3

Step-by-step explanation:

As w is a complex cube root of unity, it satisfies

w³ - 1 = 0  <=>  (w-1)(w²+w+1) = 0

But w ≠ 1, so it follows that w²+w+1 = 0.

Therefore 1+w² = -w and

1+w⁴ = 1+w³w = 1+w = -w².

We want the least positive n such that

( (1+w⁴) / (1+w²) )ⁿ = 1

<=> ( (-w²) / (-w) )ⁿ = 1

<=> wⁿ = 1

From here, we see the answer is 3.

Answer 2

Step-by-step explanation:

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