If `z = 5 + (3sqrt2) i`, find `zbarz`.

Answer

To find \( z \bar{z} \) where \( z = 5 + 3\sqrt{2} i \), we can follow these steps:

### Step 1: Identify the complex number
We have:
\[ z = 5 + 3\sqrt{2} i \]

### Step 2: Find the conjugate of \( z \)
The conjugate of a complex number \( z = x + yi \) is given by \( \bar{z} = x - yi \). Therefore, for our complex number:
\[ \bar{z} = 5 - 3\sqrt{2} i \]

### Step 3: Calculate \( z \bar{z} \)
We can use the formula:
\[ z \bar{z} = (5 + 3\sqrt{2} i)(5 - 3\sqrt{2} i) \]
Using the difference of squares:
\[ z \bar{z} = 5^2 - (3\sqrt{2} i)^2 \]

### Step 4: Compute \( 5^2 \) and \( (3\sqrt{2} i)^2 \)
Calculating \( 5^2 \):
\[ 5^2 = 25 \]
Calculating \( (3\sqrt{2} i)^2 \):
\[ (3\sqrt{2} i)^2 = (3\sqrt{2})^2 (i^2) = 9 \cdot 2 \cdot (-1) = -18 \]

### Step 5: Substitute back into the equation
Now substituting back:
\[ z \bar{z} = 25 - (-18) = 25 + 18 = 43 \]

### Final Answer
Thus, the value of \( z \bar{z} \) is:
\[ \boxed{43} \]

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