Key Concepts and Formulas

• Complex Number Form: A complex number $z$ is expressed as $z = x + yi$, where $x$ is the real part ($\operatorname{Re}(z)$) and $y$ is the imaginary part ($\operatorname{Im}(z)$), and $i = \sqrt{-1}$ (thus $i^2 = -1$).
• Conjugate of a Complex Number: The conjugate of $z = x + yi$ is denoted by $\overline{z}$ and is given by $\overline{z} = x - yi$.
• Product of a Complex Number and its Conjugate: $z \cdot \overline{z} = (x + yi)(x - yi) = x^2 + y^2$.
• Powers of $i$: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$.

Step-by-Step Solution

Step 1: Simplify the Denominator

We need to simplify the denominator, $(3 + 2i)\cdot\overline{(4 - 6i)}$. The conjugate of $4 - 6i$ is $4 + 6i$. Therefore, we have: $(3 + 2i)(4 + 6i) = 3(4) + 3(6i) + 2i(4) + 2i(6i)$ $= 12 + 18i + 8i + 12i^2$ Since $i^2 = -1$, we have: $= 12 + 26i - 12 = 26i$ So, the denominator simplifies to $26i$.

Step 2: Simplify the Numerator

We need to simplify the numerator, $(1 + 2i)^8 \cdot (1 - 2i)^2$. We can rewrite this as: $(1 + 2i)^8 (1 - 2i)^2 = (1 + 2i)^6 (1 + 2i)^2 (1 - 2i)^2 = (1 + 2i)^6 [(1 + 2i)(1 - 2i)]^2$ Now, $(1 + 2i)(1 - 2i) = 1^2 + 2^2 = 1 + 4 = 5$. So the numerator becomes: $(1 + 2i)^6 \cdot 5^2 = 25 (1 + 2i)^6$ We can rewrite $(1 + 2i)^6$ as $[(1 + 2i)^2]^3$. First calculate $(1 + 2i)^2$: $(1 + 2i)^2 = 1^2 + 2(1)(2i) + (2i)^2 = 1 + 4i + 4i^2 = 1 + 4i - 4 = -3 + 4i$ Now, we have: $25(-3 + 4i)^3$ We need to calculate $(-3 + 4i)^3$. $(-3 + 4i)^3 = (-3)^3 + 3(-3)^2(4i) + 3(-3)(4i)^2 + (4i)^3$ $= -27 + 3(9)(4i) + 3(-3)(16i^2) + 64i^3$ $= -27 + 108i - 144i^2 + 64i^3$ Since $i^2 = -1$ and $i^3 = -i$, we have: $= -27 + 108i + 144 - 64i = 117 + 44i$ Therefore, the numerator is: $25(117 + 44i) = 2925 + 1100i$

Step 3: Combine and Rationalize

Now we have the simplified numerator and denominator: $Z = \frac{2925 + 1100i}{26i}$ To rationalize the denominator, we multiply both the numerator and the denominator by $-i$: $Z = \frac{2925 + 1100i}{26i} \cdot \frac{-i}{-i} = \frac{-2925i - 1100i^2}{-26i^2} = \frac{-2925i + 1100}{26} = \frac{1100 - 2925i}{26}$ $Z = \frac{1100}{26} - \frac{2925}{26}i$

Step 4: Identify the Real Part

The real part of $Z$ is $\frac{1100}{26}$. We can simplify this fraction by dividing both numerator and denominator by 2: $\frac{1100}{26} = \frac{550}{13}$ Therefore, the real part of the complex number is $\frac{550}{13}$.

Common Mistakes & Tips

• Be careful with signs, especially when dealing with powers of $i$ and binomial expansions.
• Always simplify as much as possible at each step to avoid carrying large numbers through the calculations.
• Remember to rationalize the denominator before identifying the real part.

Summary

By simplifying the numerator and denominator separately, rationalizing the denominator, and then identifying the real part, we found the real part of the given complex number to be $\frac{550}{13}$.

The final answer is $\boxed{\frac{550}{13}}$, which corresponds to option (A).