The smallest positive integer for which , where i = √-1, is:

Concept:

Complex Numbers:

• A complex number is a number of the form a + ib, where a and b are real numbers and i is the complex unit defined by i = √-1.
• i² = -1, i³ = -i, i⁴ = 1 etc.
• In general, i^{4n + 1} = i, i4n + 2 = -1, i4n + 3 = -i, i4n = 1.

• For a complex number z = a + ib, conjugate of z is z̅ = a - ib.

Calculation:

Rationalizing the complex number , by multiplying and dividing by the conjugate of the denominator, we get:

 = -i.

Now, .

⇒ 

⇒ (-i)ⁿ for n = 2.

(-i)² = (-1)2 × (i)2 = 1 × -1 = -1.

∴  n = 2