If [(1+i)/(1-i)]ᵐ = 1, then find the least positive integral value of m

Solution:

We will convert (1+i)/(1-i) into the form a + ib by rationalizing the denominator.

(1+i)/(1-i) = (1+i)/(1-i)· (1+i)/(1+i)

= (1 + 2i + i²) / (1 - i²)

= (2i) / (2)   [because i² = -1]

= i

Substitute this in the given equation, we get

i^m = 1

We know that the power of iota is 1 only when m is a multiple of 4. Also, the problem is looking for positive values of m.

Hence, m = 4k, where k is any integer

NCERT Solutions Class 11 Maths Chapter 5 Exercise ME Question 20

If [(1+i)/(1-i)]ᵐ = 1, then find the least positive integral value of m

Summary:

For [(1+i)/(1-i)]^m = 1, m = 4k where k is any integer