if z = (8+6j)/(3-3j) express z + (1/z) in the form of x+yj

We are tasked with finding \( z + \frac{1}{z} \) in the form \( x + yj \), where \( z = \frac{8 + 6j}{3 - 3j} \).

Let's break this down step by step.

### Step 1: Simplify \( z = \frac{8 + 6j}{3 - 3j} \)
To simplify \( z \), multiply both the numerator and the denominator by the conjugate of the denominator:

\[
z = \frac{(8 + 6j)(3 + 3j)}{(3 - 3j)(3 + 3j)}
\]

Simplifying the denominator:
\[
(3 - 3j)(3 + 3j) = 9 + 9j - 9j - 9j^2 = 9 + 9 = 18
\]

Simplifying the numerator using the distributive property:
\[
(8 + 6j)(3 + 3j) = 8(3) + 8(3j) + 6j(3) + 6j(3j) = 24 + 24j + 18j + 18j^2
\]
Since \( j^2 = -1 \), the expression becomes:
\[
24 + 24j + 18j + 18(-1) = 24 + 42j - 18 = 6 + 42j
\]

So, we now have:
\[
z = \frac{6 + 42j}{18} = \frac{6}{18} + \frac{42j}{18} = \frac{1}{3} + \frac{7j}{3}
\]

Thus:
\[
z = \frac{1}{3} + \frac{7j}{3}
\]

### Step 2: Find \( \frac{1}{z} \)
To find \( \frac{1}{z} \), we use the formula for the inverse of a complex number:
\[
\frac{1}{z} = \frac{\text{conjugate of } z}{|z|^2}
\]
First, find the conjugate of \( z = \frac{1}{3} + \frac{7j}{3} \), which is \( \frac{1}{3} - \frac{7j}{3} \).

Now, find \( |z|^2 \):
\[
|z|^2 = \left( \frac{1}{3} \right)^2 + \left( \frac{7}{3} \right)^2 = \frac{1}{9} + \frac{49}{9} = \frac{50}{9}
\]

Thus:
\[
\frac{1}{z} = \frac{\frac{1}{3} - \frac{7j}{3}}{\frac{50}{9}} = \frac{9}{50} \left( \frac{1}{3} - \frac{7j}{3} \right)
\]
\[
\frac{1}{z} = \frac{9}{50} \times \frac{1}{3} - \frac{9}{50} \times \frac{7j}{3} = \frac{3}{50} - \frac{21j}{50}
\]

### Step 3: Find \( z + \frac{1}{z} \)
Now, sum \( z = \frac{1}{3} + \frac{7j}{3} \) and \( \frac{1}{z} = \frac{3}{50} - \frac{21j}{50} \):
\[
z + \frac{1}{z} = \left( \frac{1}{3} + \frac{3}{50} \right) + \left( \frac{7j}{3} - \frac{21j}{50} \right)
\]

First, calculate the real part:
\[
\frac{1}{3} + \frac{3}{50} = \frac{50}{150} + \frac{9}{150} = \frac{59}{150}
\]

Next, calculate the imaginary part:
\[
\frac{7}{3} - \frac{21}{50} = \frac{350}{150} - \frac{63}{150} = \frac{287}{150}
\]

Thus:
\[
z + \frac{1}{z} = \frac{59}{150} + \frac{287j}{150}
\]

### Final Answer:
\[
z + \frac{1}{z} = \frac{59}{150} + \frac{287j}{150}
\]

This is the required expression in the form \( x + yj \).

Would you like further details or have any questions?

Here are some related questions:
1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate when simplifying?
3. Can complex numbers be represented graphically?
4. How do you calculate the magnitude of a complex number?
5. What are the properties of complex division?

**Tip:** Always express the denominator in terms of real numbers when simplifying complex fractions.

Learn how to express z + 1/z in the form x + yj where z = (8 + 6j) / (3 - 3j). This solution involves simplifying complex numbers, using conjugates, and applying complex division. Step-by-step breakdown suitable for students in Grades 11-12.

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