Imaginary roots (2)

You first need to find the angle $x_{1}$ and $x_{a}$ are at compared to the positive real-axis on the complex plane.

$x_{1}$ is at $0$ radians to the positive real-axis as it consists of only positive real numbers.

$x_{a}$ can be calculated by using SOHCAHTOA.

Let's make the imaginary part of $x_{a}$ the opposite and the real part the adjacent on a right-angled triangle with angle $\theta$ .

$Adj = \frac{3}{2}$ $Opp = \frac{\sqrt{3}}{2}$

The opposite is positive since we're working with lengths which can only be positive.

Using trigonometry we get $\tan{\theta} = \frac{Opp}{Adj}$ which when re-arranged and solved gives us:

$\theta = \tan^{-1}{\frac{Opp}{Adj}}$ $\theta = \tan^{-1}{\frac{\sqrt{3}}{3}}$ $\theta = \frac{\pi}{6}$

Now that we have $\theta$ we find the difference between the two angles, so we can determine the value of $a$ .

$\frac{\pi}{6} - 0 = \frac{\pi}{6}$

We then divide $2\pi$ by the difference between the two angles to get $a$ .

$\frac{2\pi}{\frac{\pi}{6}} = a$ $\frac{12\pi}{\pi} = a$ $a = 12$

Finally we take one of the roots and put it to the power of $a$ to get $n$ , we'll choose $x_{1}$ since its a real number.

${x_{1}}^{12} = n$ ${\sqrt{3}}^{12} = n$ $3^{6} = n$ $\boxed{n = 729}$