Calculate value of a real number, considering "n" as a natural number

Question

How could I calculate the value of the real number:

$$ (1 +i \sqrt{3})^n + (1 - i \sqrt{3})^n $$

...considering $n$ as a natural number and $i$ as the imaginary unit.

Answer 1

Hint: $$ \bar{z}^n = \overline{z^n} \quad z + \bar{z} = 2 \mbox{Re}(z) $$

Answer 2

Hint: $$ \bigg({\frac{1 \pm i \sqrt{3}}2}\bigg)^3 = -1 $$

Answer 3

Hint: We are looking at $$2^n\left(\left(\frac{1+\sqrt{3}i}{2}\right)^n+\left(\frac{1-\sqrt{3}i}{2}\right)^n\right).$$ Inside we have sixth roots of unity, which cycle nicely.

Remark: I prefer to work with the complex exponential, but one could use trigonometric. Note that $$1+\sqrt{3}\,i=2(\cos\theta+i\sin\theta),$$ where $\theta=\frac{\pi}{3}$. Taking the $n$-th power, and using the de Moivre formula, we get $$(1+\sqrt{3}\,i)^n=2^n(\cos n\theta+i\sin n\theta).$$ Similarly, $$(1-\sqrt{3}\,i)^n=2^n(\cos n\theta-i\sin n\theta).$$ Add.