We are tasked with solving the equation:

\[
z^7 = 0.5 \times (1 + \sqrt{3}i)
\]

This is a complex equation, so let's proceed step by step.

### Step 1: Convert the right-hand side into polar form

The complex number \( 0.5 \times (1 + \sqrt{3}i) \) needs to be written in polar form \( re^{i\theta} \), where \( r \) is the modulus (magnitude) and \( \theta \) is the argument (angle).

1. **Magnitude \( r \)**:
   The magnitude of \( 0.5 \times (1 + \sqrt{3}i) \) is calculated as:
   \[
   r = 0.5 \times \sqrt{1^2 + (\sqrt{3})^2} = 0.5 \times \sqrt{1 + 3} = 0.5 \times \sqrt{4} = 0.5 \times 2 = 1
   \]

2. **Argument \( \theta \)**:
   The argument \( \theta \) is the angle in the complex plane for \( 1 + \sqrt{3}i \). Using the formula:
   \[
   \theta = \text{arg}(1 + \sqrt{3}i) = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}
   \]
   Thus, the polar form of \( 0.5 \times (1 + \sqrt{3}i) \) is:
   \[
   1 \times e^{i\frac{\pi}{3}} = e^{i\frac{\pi}{3}}
   \]

### Step 2: Solve the equation \( z^7 = e^{i\frac{\pi}{3}} \)

To solve for \( z \), we take the 7th root of both sides. Using De Moivre's Theorem, the general solution for \( z \) is given by:
\[
z = e^{i\left(\frac{\frac{\pi}{3} + 2k\pi}{7}\right)}, \quad k = 0, 1, 2, \dots, 6
\]
This gives us the 7 different solutions corresponding to different values of \( k \).

### Step 3: Write out the solutions

For \( k = 0 \) to \( k = 6 \), the solutions are:
\[
z_k = e^{i\left(\frac{\pi}{21} + \frac{2k\pi}{7}\right)}, \quad k = 0, 1, 2, \dots, 6
\]

These represent the 7 distinct complex roots of the equation.

Would you like a detailed explanation of each solution or how these roots behave on the complex plane?

### Additional Related Questions:
1. What is De Moivre's Theorem and how does it apply to roots of complex numbers?
2. How do we convert complex numbers from Cartesian to polar form?
3. What are the geometrical interpretations of complex roots on the unit circle?
4. How do multiple roots of an equation like \( z^7 \) distribute on the complex plane?
5. Can you derive the magnitude and argument of any general complex number?

### Tip:
Remember, the roots of any complex number \( z^n = w \) are spaced evenly on the unit circle when the modulus of \( w \) is 1.

Explore the solution to z^7 = 0.5*(1+√3i) by converting to polar form and applying De Moivre's Theorem. This guide provides a step-by-step analysis of complex roots, suitable for Grades 11-12 students.

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