https://file.aiquickdraw.com/mathbot/file/17259346340728mry4a4g.jpg

The given expression in the image is:

\[
\left(\frac{1 + \sqrt{-3}}{2}\right)^6 + \left(\frac{1 - \sqrt{-3}}{2}\right)^9
\]

This expression involves complex numbers due to the square root of \(-3\), which can be simplified using imaginary numbers (\(i\)), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-3} = i\sqrt{3}\).

### Rewriting the expression:
\[
\left(\frac{1 + i\sqrt{3}}{2}\right)^6 + \left(\frac{1 - i\sqrt{3}}{2}\right)^9
\]

Next, we can recognize that the terms inside the parentheses resemble the cube roots of unity. These cube roots are related to the complex numbers \( \omega \) and \( \omega^2 \), where:

\[
\omega = \frac{1 + i\sqrt{3}}{2}, \quad \omega^2 = \frac{1 - i\sqrt{3}}{2}
\]

Thus, the expression can be interpreted as involving powers of the cube roots of unity.

#### Cube roots of unity properties:
\[
\omega^3 = 1, \quad \omega^6 = 1, \quad \omega^9 = 1
\]

Since both powers of \(\omega\) and \(\omega^2\) are integers that are multiples of 3, the expression simplifies as follows:

\[
\omega^6 = 1 \quad \text{and} \quad (\omega^2)^9 = 1
\]

Thus, the entire expression becomes:

\[
1 + 1 = 2
\]

The simplified result is:
\[
\boxed{2}
\]

Would you like more details or have any questions? Here are five related questions you can explore:
1. What are the cube roots of unity, and how are they used in complex number calculations?
2. How do imaginary numbers like \(i\) and \(\sqrt{-1}\) work in mathematical expressions?
3. Can we generalize this result for other powers of complex roots?
4. What is the geometric interpretation of complex numbers and roots of unity on the complex plane?
5. How do De Moivre’s theorem and Euler’s formula relate to powers of complex numbers?

**Tip:** Understanding the cube roots of unity helps in simplifying many complex expressions, especially those involving periodicity.

Simplify the expression: ( (1 + √-3) / 2 )^6 + ( (1 - √-3) / 2 )^9

This problem involves simplifying a complex expression with cube roots of unity and imaginary numbers. Learn step-by-step how to apply the properties of cube roots and simplify the expression to a final answer of 2. Suitable for advanced high school mathematics students.

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