1. **State the problem:** Simplify the complex expression $$\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^{12}$$. 2. **Recall the formula and rules:** To simplify powers of complex numbers, it's easiest to convert the fraction to polar form using Euler's formula: $$re^{i\theta}$$, where $r$ is the magnitude and $\theta$ is the argument (angle). 3. **Calculate the magnitude $r$ of the fraction:** $$r = \frac{|1-\sqrt{3}i|}{|1+\sqrt{3}i|} = \frac{\sqrt{1^2 + (\sqrt{3})^2}}{\sqrt{1^2 + (\sqrt{3})^2}} = \frac{\sqrt{1+3}}{\sqrt{1+3}} = \frac{2}{2} = 1$$ 4. **Calculate the argument $\theta$ of numerator and denominator:** - Numerator angle: $$\theta_1 = \arctan\left(\frac{-\sqrt{3}}{1}\right) = -\frac{\pi}{3}$$ - Denominator angle: $$\theta_2 = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$$ 5. **Argument of the fraction:** $$\theta = \theta_1 - \theta_2 = -\frac{\pi}{3} - \frac{\pi}{3} = -\frac{2\pi}{3}$$ 6. **Express the fraction in polar form:** $$1 \cdot e^{i(-2\pi/3)} = e^{-i2\pi/3}$$ 7. **Raise to the 12th power:** $$\left(e^{-i2\pi/3}\right)^{12} = e^{-i2\pi/3 \times 12} = e^{-i8\pi}$$ 8. **Simplify the exponent:** Since $e^{-i8\pi} = e^{-i2\pi \times 4} = (e^{-i2\pi})^4 = 1^4 = 1$ **Final answer:** $$\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^{12} = 1$$