We are given:

$z = \frac{1}{3} + \frac{\sqrt{3}}{2} i$

and need to solve for $\overline{z}^4$, where $\overline{z}$ is the complex conjugate of $z$.

Step 1: Find the complex conjugate of $z$

The complex conjugate of $z = \frac{1}{3} + \frac{\sqrt{3}}{2} i$ is obtained by changing the sign of the imaginary part. Therefore:

$\overline{z} = \frac{1}{3} - \frac{\sqrt{3}}{2} i$

Step 2: Express $\overline{z}$ in polar form

To simplify the calculation of $\overline{z}^4$, it's helpful to convert $\overline{z}$ to polar form. The polar form of a complex number is given by:

$\overline{z} = r \left( \cos \theta + i \sin \theta \right)$

where:

• $r$ is the modulus of $\overline{z}$,
• $\theta$ is the argument (angle) of $\overline{z}$.

Modulus $r$:

The modulus $r$ of $\overline{z}$ is:

$r = \left| \overline{z} \right| = \sqrt{\left( \frac{1}{3} \right)^2 + \left( \frac{-\sqrt{3}}{2} \right)^2}$

r = \sqrt{\frac{1}{9} + \frac{3}{4}} = \sqrt{\frac{1 + 6.75}{9}} = \sqrt{\frac{7.75}{9}} = \frac{\sqrt{31}{6}}