The multiplicative inverse of `(3 + 2i)^2` is :

Options

A. `(12)/(169) - (5i)/(169)`
B. `5/(169) - (12i)/(169)`
C. `5/13 - (12i)/(13)`
D. none of these

Answer

To find the multiplicative inverse of \((3 + 2i)^2\), we will follow these steps:

### Step 1: Calculate \((3 + 2i)^2\)
We start by expanding the expression:
\[ (3 + 2i)^2 = 3^2 + 2 \cdot 3 \cdot 2i + (2i)^2 \]
Calculating each term:
- \(3^2 = 9\)
- \(2 \cdot 3 \cdot 2i = 12i\)
- \((2i)^2 = 4i^2 = 4(-1) = -4\)

Now, combine these results:
\[ (3 + 2i)^2 = 9 + 12i - 4 = 5 + 12i \]

### Step 2: Find the multiplicative inverse of \(5 + 12i\)
The multiplicative inverse of a complex number \(z = a + bi\) is given by:
\[ z^{-1} = \frac{1}{a + bi} = \frac{1}{5 + 12i} \]

To find this, we multiply the numerator and denominator by the conjugate of the denominator:
\[ z^{-1} = \frac{1 \cdot (5 - 12i)}{(5 + 12i)(5 - 12i)} \]

### Step 3: Simplify the denominator
Calculating the denominator:
\[ (5 + 12i)(5 - 12i) = 5^2 - (12i)^2 = 25 - 144(-1) = 25 + 144 = 169 \]

### Step 4: Write the inverse
Now substituting back:
\[ z^{-1} = \frac{5 - 12i}{169} \]

### Step 5: Separate into real and imaginary parts
This can be expressed as:
\[ z^{-1} = \frac{5}{169} - \frac{12i}{169} \]

### Conclusion
Thus, the multiplicative inverse of \((3 + 2i)^2\) is:
\[ \frac{5}{169} - \frac{12i}{169} \]

### Final Answer
The correct option is the second option: \(\frac{5}{169} - \frac{12i}{169}\).

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