Question

If \(z = 2 + 3i\), then \(z^5 + (\overline{z})^5\) is equal to:

• A. 244
• B. 224
• C. 245
• D. 265

Answer 1

The Correct Option is A

To solve the problem of finding the value of \(z^5 + (\overline{z})^5\), we start with the given complex number:

\(z = 2 + 3i\)

First, calculate \(z^5\). To do this, use the binomial theorem:

Step 1: Calculate \(z^2 = (2 + 3i)^2\)

\(z^2 = (2 + 3i)^2 = 4 + 12i + 9i^2\)

Since \(i^2 = -1\), we have:

\(z^2 = 4 + 12i - 9 = -5 + 12i\)

Step 2: Calculate \(z^3 = z \cdot z^2 = (2 + 3i)(-5 + 12i)\)

\(z^3 = 2(-5) + 2(12i) + 3i(-5) + 3i(12i) = -10 + 24i - 15i + 36i^2\)

\(z^3 = -10 + 9i - 36 = -46 + 9i\)

Step 3: Calculate \(z^4 = z \cdot z^3 = (2 + 3i)(-46 + 9i)\)

\(z^4 = 2(-46) + 2(9i) + 3i(-46) + 3i(9i) = -92 + 18i - 138i + 27i^2\)

\(z^4 = -92 - 120i - 27 = -119 - 120i\)

Step 4: Calculate \(z^5 = z \cdot z^4 = (2 + 3i)(-119 - 120i)\)

\(z^5 = 2(-119) + 2(-120i) + 3i(-119) + 3i(-120i) = -238 - 240i - 357i - 360i^2\)

\(z^5 = -238 - 597i + 360 = 122 - 597i\)

Next, find the complex conjugate of \(z\), which is \(\overline{z} = 2 - 3i\) and calculate \((\overline{z})^5\).

Step 5: Use the symmetry property of magnitudes:

The magnitude (or modulus) of \(z\) and \(\overline{z}\) are equal, which implies

\((z \cdot \overline{z})^5 = |z|^{10}\), and due to conjugation property it is equal to \(z^5 + (\overline{z})^5\).

Given:

\(z \cdot \overline{z} = (2+3i)(2-3i) = 4 + 9 = 13\)

Thus, \(|z|^{10} = 13^5 = 371293\). However, due to algebraic symmetry involving odd powers, we need to find \(z^5 + (\overline{z})^5\).

Through actual computation and given options, notice simplifying calculation errors and proceed using symmetry interpretation:

\(z^5 = 122 - 597i\) explains that symmetry and magnitude ensure real part doubling from originally similar algebraic progression evaluations leading \(122 + 122\) to match 244, aligning with symmetry. Calculation traps require caution!

Conclusion: The correct answer is 244.