Evaluate $(\sqrt{3}+i)^{14}+(\sqrt{3}-i)^{14}$ Evaluate $(\sqrt{3}+i)^{14}+(\sqrt{3}-i)^{14}$
I tried by using the De'moivers theorem but I didn't get proper value I get a mess value 4..but I am not sure about the answer can anyone please tell me
 A: First, note that $z=\sqrt 3+i=2e^{i\pi/6}$. Then $z^{14}=2^{14}e^{14i\pi/6}=2^{14}e^{i\pi/3}$.
Now, $z^{14}+\bar z^{14}=2\Re(z^{14})=2\cdot2^{14}\cos(\pi/3)=2^{14}=16384$
A: For a different sort of computation:
Let $$a_n=(\sqrt 3 +i)^{2n}+(\sqrt 3-i)^{2n}$$
We note that $\alpha = (\sqrt 3+i)$ satisfies $x^4-4x^2+16$ from which it follows that $\alpha^2$ satisfies $x^2-4x+16$.   It follows that we have the recursion $$a_n=4a_{n-1}-16a_{n-2}$$  As it is clear that $$a_0=2\quad \& \quad a_1=4$$  we can easily compute $$a_7=16384$$
Side note:  somewhat curiously, it is not obvious from this point of view that $a_n$ is always a power of $2$, at least up to sign.  Indeed, the relevant powers of $2$, namely $\{1,2,4,7,8,10,13,\cdots\}$, follow a (slightly) unexpected pattern.
A: First of all, try to factorise by $2^{14}$ :).
$(\sqrt{3} + i)^{14} + (\sqrt{3} - i)^{14} = 2^{14}(\dfrac{\sqrt{3}}{2} + \dfrac{i}{2})^{14} + 2^{14}(\dfrac{\sqrt{3}}{2} - \dfrac{i}{2})^{14} = 2^{14}\left((\dfrac{\sqrt{3}}{2} + \dfrac{i}{2})^{14} + (\dfrac{\sqrt{3}}{2} - \dfrac{i}{2})^{14}\right)$
Now, use the fact that: $\dfrac{\sqrt{3}}{2} + \dfrac{i}{2} = \cos(\pi/6) + i\sin(\pi/6) = \exp(i\pi/6)$.
And similarly: $\dfrac{\sqrt{3}}{2} - \dfrac{i}{2} = \cos(-\pi/6) + i\sin(-\pi/6) = \exp(-i\pi/6)$.
I think you'll be able to finish from there, by using De Moivre's theorem :)
A: Observe that
$$
(\sqrt{3} + i)^{14} = ((\sqrt{3} + i)^2)^7
$$
Also, $ (\sqrt{3} + i)^2 = 2 + i 2\sqrt{3} = 2 (2 e^{j\theta}) = 4 e^{j\theta}$, where $ \theta = \pi / 3 $.
Similarly, for the second term we have that $ (\sqrt{3} - i)^2 = 4 e^{-j\theta} $.
You problem reduces to finding 
$$ 
M = 4^7 \cdot e^{j 7\theta} + 4^7 \cdot e^{-j 7\theta}  
= 4^7 \cdot 2 \cdot (\frac{e^{j 7\theta} + e^{-j 7\theta}}{2})  
= 4^7 \cdot \cos(7\theta)
= 4^7 \cdot 1
= 16384
$$ 
