To understand the formula in the image, let's break down the mathematics step-by-step based on your uploaded content:

1. Expression for $Z^5$: The formula appears to describe the relationship between a complex number $Z$ in polar form and its magnitude and argument.

2. Polar Representation: Any complex number $Z$ can be represented in polar form as: $Z = r (\cos \theta + i \sin \theta) = r e^{i\theta},$ where $r$ is the modulus ($|Z|$) and $\theta$ is the argument ($\arg(Z)$).

3. Equation Given: From the image, $Z^5 = Z\left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)$:

   • $\frac{1}{2} + i\frac{\sqrt{3}}{2}$ is a known complex number in polar form with $r = 1$ and argument $\frac{\pi}{3}$.

4. Key Observation: To satisfy $Z^5 = Z \cdot (\frac{1}{2} + i\frac{\sqrt{3}}{2})$, the modulus $r$ of $Z$ must be consistent after raising to the 5th power: $r^5 = r.$ This leads to the condition that $r = 1$ (since the modulus of $Z$ is normalized for polar rotations).

5. Verification: The square root term used, $\sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$, ensures the modulus normalization.

Why This Formula is Used:

The formula stems from the fact that for complex numbers:

• Rotations (like multiplying by $\frac{1}{2} + i\frac{\sqrt{3}}{2}$) occur on the unit circle, meaning the modulus $r$ remains $1$ after successive powers.
• The argument $\theta$ of $Z$ rotates by multiples of $\frac{\pi}{3}$ due to the multiplication factor.

Do you want me to explain in more depth, such as working through the derivation or extending to examples?

Related Questions:

1. How do you convert a complex number from rectangular to polar form?
2. What is the geometric interpretation of multiplying complex numbers?
3. Why does the unit circle play a significant role in these transformations?
4. How does De Moivre's theorem apply to powers of complex numbers?
5. What are the practical applications of this formula?

Tip:

Remember, multiplying complex numbers in polar form corresponds to adding their arguments