Key Concepts and Formulas

• Complex Conjugate: If $z = a + bi$, where $a$ and $b$ are real numbers, then its complex conjugate is $\bar{z} = a - bi$.
• Conjugate of a Power: For any complex number $z$ and integer $n$, $\overline{z^n} = (\bar{z})^n$.
• Sum of a Complex Number and its Conjugate: $z + \bar{z} = 2\text{Re}(z)$, where $\text{Re}(z)$ denotes the real part of $z$.

Step-by-Step Solution

1. Simplify the expression using conjugate properties

We are given the expression $z^5 + (\bar{z})^5$. Our goal is to simplify this expression using the properties of complex conjugates.

Using the property $\overline{z^n} = (\bar{z})^n$, we can rewrite $(\bar{z})^5$ as $\overline{z^5}$. $z^5 + (\bar{z})^5 = z^5 + \overline{z^5}$

Now, using the property $z + \bar{z} = 2\text{Re}(z)$, we can rewrite the expression as: $z^5 + \overline{z^5} = 2\text{Re}(z^5)$

Thus, we only need to find the real part of $z^5$ and multiply it by 2.

2. Calculate $z^2$

We are given $z = 2 + 3i$. We need to calculate $z^2$ to find higher powers of $z$. $z^2 = (2 + 3i)^2$

Expanding the square: $z^2 = (2)^2 + 2(2)(3i) + (3i)^2$ $z^2 = 4 + 12i + 9i^2$

Since $i^2 = -1$: $z^2 = 4 + 12i - 9$ $z^2 = -5 + 12i$

3. Calculate $z^4$

Now, we calculate $z^4$ by squaring $z^2$: $z^4 = (z^2)^2 = (-5 + 12i)^2$

Expanding the square: $z^4 = (-5)^2 + 2(-5)(12i) + (12i)^2$ $z^4 = 25 - 120i + 144i^2$

Since $i^2 = -1$: $z^4 = 25 - 120i - 144$ $z^4 = -119 - 120i$

4. Calculate $z^5$

Next, we calculate $z^5$ by multiplying $z^4$ and $z$: $z^5 = z^4 \cdot z = (-119 - 120i)(2 + 3i)$

Expanding the product: $z^5 = (-119)(2) + (-119)(3i) + (-120i)(2) + (-120i)(3i)$ $z^5 = -238 - 357i - 240i - 360i^2$

Since $i^2 = -1$: $z^5 = -238 - 357i - 240i + 360$ $z^5 = (-238 + 360) + (-357 - 240)i$ $z^5 = 122 - 597i$

5. Calculate $2\text{Re}(z^5)$

We have $z^5 = 122 - 597i$. Therefore, the real part of $z^5$ is: $\text{Re}(z^5) = 122$

Finally, we calculate $2\text{Re}(z^5)$: $2\text{Re}(z^5) = 2(122) = 244$

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when expanding and simplifying expressions involving complex numbers, especially when dealing with $i^2 = -1$.
• Iterative Calculation: Break down higher powers into smaller, manageable steps to minimize errors.
• Conjugate Properties: Remember to use the properties of conjugates to simplify the expressions, especially when dealing with sums and powers.

Summary

We used the properties of complex conjugates to simplify the expression $z^5 + (\bar{z})^5$ to $2\text{Re}(z^5)$. We then calculated $z^5$ by iteratively finding $z^2$, $z^4$, and finally $z^5$. The real part of $z^5$ was found to be 122, and therefore, $z^5 + (\bar{z})^5 = 2(122) = 244$.

The final answer is \boxed{244}, which corresponds to option (A).