Given the complex number $z$ satisfying the equation

$z - \frac{1}{z} = 4 + 4i,$

we need to find the modulus (or length) of $z + \frac{1}{z}$.

Step 1: Let $z = a + bi$, where $a$ and $b$ are real numbers.

The reciprocal $\frac{1}{z}$ can be expressed as:

$\frac{1}{z} = \frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2}$

Thus, the equation $z - \frac{1}{z} = 4 + 4i$ becomes:

$(a + bi) - \frac{a - bi}{a^2 + b^2} = 4 + 4i$

Step 2: Separate the equation into real and imaginary parts.

Let's express the equation as:

$a + bi - \left(\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\right) = 4 + 4i$

Equating real and imaginary parts:

Real part:

$a - \frac{a}{a^2 + b^2} = 4$

Imaginary part:

$b + \frac{b}{a^2 + b^2} = 4$

Step 3: Simplify and solve for $a$ and $b$.

Real part equation:

$a\left(1 - \frac{1}{a^2 + b^2}\right) = 4$

This simplifies to:

$a\left(\frac{a^2 + b^2 - 1}{a^2 + b^2}\right) = 4$

Imaginary part equation:

$b\left(1 + \frac{1}{a^2 + b^2}\right) = 4$

This simplifies to:

$b\left(\frac{a^2 + b^2 + 1}{a^2 + b^2}\right) = 4$

Step 4: Find $z + \frac{1}{z}$.

Now, we calculate $z + \frac{1}{z}$:

$z + \frac{1}{z} = (a + bi) + \left(\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\right)$

$= \left(a + \frac{a}{a^2 + b^2}\right) + \left(b - \frac{b}{a^2 + b^2}\right)i$

Step 5: Calculate the modulus of $z + \frac{1}{z}$.

The modulus is given by:

$\left| z + \frac{1}{z} \right| = \sqrt{\left(a + \frac{a}{a^2 + b^2}\right)^2 + \left(b - \frac{b}{a^2 + b^2}\right)^2}$

We know from the given condition:

$\left| z - \frac{1}{z} \right|^2 = |4 + 4i|^2 = 4^2 + 4^2 = 32$

Thus, the modulus of $z + \frac{1}{z}$ is also $\sqrt{32} = 4\sqrt{2}$.

Final Answer:

The modulus of $z + \frac{1}{z}$ is $4\sqrt{2}$.

Would you like to see more details, or have any questions? Here are 5 related questions to consider:

1. What is the general method to find the modulus of a complex number?
2. How do you find the modulus of a complex expression like $z - \frac{1}{z}$?
3. Can you derive a general formula for $|z + \frac{1}{z}|$ in terms of $|z|$?
4. What is the geometric interpretation of the modulus of a complex number?
5. How does the modulus change when a complex number is inverted (i.e., $\frac{1}{z}$)?

Tip: When solving complex equations, separating real and imaginary parts is often key to finding solutions.