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The problem involves a complex number \(\alpha = x + iy\), where \(x, y \in \mathbb{R}\) and \(i = \sqrt{-1}\). We are asked to find the value of the following expression:

\[
\frac{(x+iy)^2 (x+iy)^3}{(x-iy)^4 (x-iy)^5} + \frac{(x^2)}{|\alpha|}
\]

### Step-by-step breakdown:

#### 1. Simplifying the first term:
The first part of the expression is:

\[
\frac{(x+iy)^2 (x+iy)^3}{(x-iy)^4 (x-iy)^5} 
\]

This simplifies by combining the exponents:
\[
= \frac{(x+iy)^{2+3}}{(x-iy)^{4+5}} = \frac{(x+iy)^5}{(x-iy)^9}
\]

#### 2. Expressing in polar form:
For complex numbers \(x + iy\) and \(x - iy\), we can use the polar form. The polar form of \(\alpha = x + iy\) is:

\[
\alpha = r e^{i\theta}, \quad \text{where} \ r = |\alpha| = \sqrt{x^2 + y^2}, \quad \theta = \arg(\alpha)
\]

Similarly, the polar form of \(x - iy\) is:
\[
x - iy = r e^{-i\theta}
\]

Thus, the expression becomes:

\[
\frac{r^5 e^{i5\theta}}{r^9 e^{-i9\theta}} = \frac{e^{i5\theta}}{r^4 e^{-i9\theta}} = \frac{e^{i5\theta}}{e^{-i9\theta}} \cdot \frac{1}{r^4}
= \frac{e^{i5\theta} e^{i9\theta}}{r^4} = \frac{e^{i14\theta}}{r^4}
\]

#### 3. Simplifying the second term:
The second part of the expression is:

\[
\frac{x^2}{|\alpha|}
\]

Since \( |\alpha| = \sqrt{x^2 + y^2} \), the second term becomes:

\[
\frac{x^2}{\sqrt{x^2 + y^2}}
\]

#### 4. Final expression:
Now we have the two terms:
\[
\frac{e^{i14\theta}}{r^4} + \frac{x^2}{\sqrt{x^2 + y^2}}
\]

This combines the simplified complex exponential term and the real-valued term.

If you'd like further details on how to handle the exponential or any other step, feel free to ask!

### Follow-up questions:
1. How can we express \(e^{i\theta}\) in terms of trigonometric functions?
2. What is the significance of polar form in simplifying complex expressions?
3. How do we calculate the magnitude \(|\alpha|\) for a given complex number?
4. Can the result be further simplified if specific values for \(x\) and \(y\) are provided?
5. How does Euler's formula apply in this context?

**Tip:** When simplifying expressions involving powers of complex numbers, switching to polar form often makes the algebra more straightforward.

Consider a non-zero complex number α = x + iy, where x, y ∈ R and i = √-1. Then the value of [(x + iy)^2 (x + iy)^3] / [(x - iy)^4 (x - iy)^5] + (x^2) / |α| is?

Solve a complex number expression with exponents and polar forms, including the use of Euler's formula. Learn how to simplify complex numbers and apply key algebraic concepts. Suitable for students in Grades 11-12.

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