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The image you uploaded contains a complex number expression, which involves operations with complex numbers (including imaginary unit \( i \), where \( i^2 = -1 \)).

### Let's break it down step by step:

1. **Numerator** of the first term: 
   \[
   z_2^2 + z_2^3
   \]
   This looks like a sum of powers of a complex number \( z_2 \).

2. **Denominator** of the first term:
   \[
   2z_2 + 8z_1
   \]
   This is the sum of a linear combination of \( z_2 \) and \( z_1 \) (two different complex numbers).

3. **Numerator** of the second term: 
   \[
   (4 + 4i + i^2) + (-7 - 24i)(3 - 4i)
   \]
   The first part is a complex number expression, while the second part involves multiplying two complex numbers.

4. **Denominator** of the second term: 
   \[
   6 - 8i + 16 + 8i
   \]
   Combining real and imaginary parts.

5. **Third term (complex fraction)**:
   \[
   \frac{4 + 4i + i^2 - 21 + 28i - 72i + 96i^2}{22}
   \]
   This involves simplifying complex expressions both in the numerator and the denominator.

### Simplifications:

- Simplify \( i^2 = -1 \) wherever it appears.
- Perform basic arithmetic for real and imaginary parts in each expression.

Let's work through the calculation step-by-step to simplify everything. I'll start with the second term and third term as they contain clear complex operations.The simplified forms of the expressions are as follows:

1. The **numerator** of the second term simplifies to \( -114 - 40i \).
2. The **denominator** of the second term simplifies to \( 22 \).
3. The **numerator** of the third term also simplifies to \( -114 - 40i \), and its denominator is \( 22 \).

Thus, both the second and third terms simplify to the same expression:
\[
\frac{-114 - 40i}{22}
\]

We can further simplify this fraction. Let me finalize the simplification.The simplified form of both the second and third terms is:
\[
\frac{-57}{11} - \frac{20i}{11}
\]

This represents the final result for these terms.

Do you have any questions or would you like a more detailed explanation of the steps? Here are some additional related questions:

1. How can you perform arithmetic operations on complex numbers?
2. What are the rules for simplifying complex fractions?
3. How is multiplication of complex numbers performed?
4. What is the significance of the imaginary unit \( i \) and \( i^2 = -1 \)?
5. How would you represent the final result in polar form?

**Tip**: When simplifying complex numbers, always keep track of real and imaginary parts separately to avoid mistakes.

Simplify the given complex number expression, involving powers of complex numbers, and arithmetic operations with imaginary unit i, where i^2 = -1.

Step-by-step guide to simplifying a complex number expression involving powers of complex numbers and imaginary unit i. Learn to handle complex arithmetic operations and simplify fractions. Ideal for Grades 10-12.

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