The multiplicative inverse of `(6 + 5i) ^(2)` is

Options

A. `(11)/(61) - (60)/(61) I`
B. `(11)/(61) + (60)/(61) I `
C. `(9)/(61) - (60)/(61) I`
D. None

Answer

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

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

Using the formula for squaring a binomial, \( (a + b)^2 = a^2 + 2ab + b^2 \):

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

Calculating each term:

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

Combining these results:

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

### Step 2: Find the multiplicative inverse

The multiplicative inverse of a complex number \( z = a + bi \) is given by:

\[ z^{-1} = \frac{1}{z} = \frac{1}{a + bi} \]

In our case, \( z = 11 + 60i \):

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

### Step 3: Multiply by the conjugate

To eliminate the imaginary part from the denominator, multiply the numerator and denominator by the conjugate of the denominator:

\[ z^{-1} = \frac{1}{11 + 60i} \cdot \frac{11 - 60i}{11 - 60i} = \frac{11 - 60i}{(11 + 60i)(11 - 60i)} \]

### Step 4: Calculate the denominator

The denominator can be calculated using the difference of squares:

\[ (11 + 60i)(11 - 60i) = 11^2 - (60i)^2 = 121 - 3600(-1) = 121 + 3600 = 3721 \]

### Step 5: Write the result

Now substituting back into the expression for the multiplicative inverse:

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

### Final Answer

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

\[ \frac{11}{3721} - \frac{60i}{3721} \]

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