Find the multiplicative inverse of each of the following complex numbers when it exists.
`(6+ 5i)^(2)`

Answer

To find the multiplicative inverse of the complex number \( (6 + 5i)^2 \), we will follow these steps:

### Step 1: Calculate \( (6 + 5i)^2 \)

Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \):

\[ (6 + 5i)^2 = 6^2 + 2 \cdot 6 \cdot 5i + (5i)^2 \]

Calculating each term:

- \( 6^2 = 36 \)
- \( 2 \cdot 6 \cdot 5i = 60i \)
- \( (5i)^2 = 25i^2 = 25(-1) = -25 \)

Now, combine these results:

\[ (6 + 5i)^2 = 36 + 60i - 25 = 11 + 60i \]

### Step 2: Find the multiplicative inverse of \( z = 11 + 60i \)

The multiplicative inverse \( z^{-1} \) of a complex number \( z \) is given by:

\[ z^{-1} = \frac{\bar{z}}{|z|^2} \]

Where \( \bar{z} \) is the conjugate of \( z \) and \( |z| \) is the modulus of \( z \).

### Step 3: Calculate the conjugate \( \bar{z} \)

The conjugate of \( z = 11 + 60i \) is:

\[ \bar{z} = 11 - 60i \]

### Step 4: Calculate the modulus \( |z| \)

The modulus \( |z| \) is calculated as follows:

\[ |z| = \sqrt{(11)^2 + (60)^2} \]

Calculating \( |z|^2 \):

\[ |z|^2 = 11^2 + 60^2 = 121 + 3600 = 3721 \]

### Step 5: Substitute into the formula for the inverse

Now we can find the multiplicative inverse:

\[ z^{-1} = \frac{11 - 60i}{3721} \]

This can be separated into real and imaginary parts:

\[ z^{-1} = \frac{11}{3721} - \frac{60}{3721}i \]

### Final Answer

Thus, the multiplicative inverse of \( (6 + 5i)^2 \) is:

\[ z^{-1} = \frac{11}{3721} - \frac{60}{3721}i \]

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