If (√(3) + i)ⁿ = (√(3) - i)ⁿ, find the smallest positive integer value of n — JEE Mathematics

Rewrite both sides in polar form:

$$\sqrt{3} + i = 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) = 2e^{i\pi/6}$$

$$\sqrt{3} - i = 2\left(\cos\frac{\pi}{6} - i\sin\frac{\pi}{6}\right) = 2e^{-i\pi/6}$$

Substitute these into the equation:

$$(2e^{i\pi/6})^n = (2e^{-i\pi/6})^n$$

$$2^n e^{in\pi/6} = 2^n e^{-in\pi/6}$$

Divide by $2^n e^{-in\pi/6}$:

$$e^{i 2n\pi/6} = 1 \implies e^{in\pi/3} = 1$$

For $e^{in\pi/3} = 1$, the exponent must be a multiple of $2\pi i$:

$$\frac{n\pi}{3} = 2k\pi \implies n = 6k \quad (k \in \mathbb{Z})$$

The smallest positive integer $n$ occurs when $k = 1$:

$$n = 6$$

Answer: 6