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The expression given in the image involves complex numbers and uses De Moivre's Theorem. The expression is:

\[
\left(\sqrt{3} \left(\cos\left(\frac{2\pi}{9}\right) + i \sin\left(\frac{2\pi}{9}\right)\right)\right)^6
\]

To simplify using De Moivre's Theorem, recall that:

\[
(r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))
\]

For the given expression, where \( r = \sqrt{3} \) and \( \theta = \frac{2\pi}{9} \), and \( n = 6 \), the formula becomes:

\[
(\sqrt{3})^6 \left(\cos\left(6 \times \frac{2\pi}{9}\right) + i \sin\left(6 \times \frac{2\pi}{9}\right)\right)
\]

Calculating \( (\sqrt{3})^6 \) and simplifying the angles:
- \( (\sqrt{3})^6 = 3^3 = 27 \)
- \( 6 \times \frac{2\pi}{9} = \frac{12\pi}{9} = \frac{4\pi}{3} \)

Substituting back:

\[
27 \left(\cos\left(\frac{4\pi}{3}\right) + i \sin\left(\frac{4\pi}{3}\right)\right)
\]

The values of cosine and sine for \( \frac{4\pi}{3} \) are:
- \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \)
- \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \)

Thus, the final expression is:
\[
27 \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = -\frac{27}{2} - i\frac{27\sqrt{3}}{2}
\]

This is the simplified form of the given complex number expression using De Moivre's Theorem.

Do you have any further questions or need more details about this topic? Here are some follow-up questions to consider:
1. What is De Moivre's Theorem and how is it derived?
2. How does the modulus and argument of a complex number change when raised to a power?
3. What are the applications of De Moivre's Theorem?
4. Can you provide examples of how to find roots of complex numbers using De Moivre's Theorem?
5. What are Euler's formulas and how do they relate to complex numbers?

Tip: When working with complex numbers, especially in trigonometric form, always consider simplifying the angle within the cosine and sine functions to its equivalent within the range from \(0\) to \(2\pi\).

This example explains how to simplify the expression [√3(cos(2π/9) + i sin(2π/9))]^6 using De Moivre's Theorem. It covers key concepts such as complex numbers in polar form, trigonometric identities, and step-by-step simplifications. Suitable for advanced high school or college students.

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