let z=1/3 + ((square root of 3) /2)i. solve for (z̄)^4. Note z̄ is the conjugate of z

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:

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

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


Let z = 1/3 + ((square root of 3) /2)i. Solve for (z̄)^4. Note z̄ is the conjugate of z.

Learn how to solve (z̄)^4 for the complex number z = 1/3 + ((sqrt(3))/2)i. This solution involves finding the complex conjugate, converting to polar form, and using De Moivre's Theorem. Suitable for advanced high school students in Grades 11-12.

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