The multiplicative inverse of ` (3+sqrt5i)^(2)` is

Options

A. ` 1/49 - ( 3sqrt5)/98 i`
B. `1/49 + ( 3sqrt5)/98 i`
C. `4+6sqrt5i`
D. ` 4-6sqrt5i`

Answer

To find the multiplicative inverse of \( (3 + \sqrt{5}i)^2 \), we will follow these steps:

### Step 1: Calculate \( (3 + \sqrt{5}i)^2 \)
Using the formula for squaring a binomial, \( (a + b)^2 = a^2 + 2ab + b^2 \):
- Here, \( a = 3 \) and \( b = \sqrt{5}i \).
- Calculate \( a^2 = 3^2 = 9 \).
- Calculate \( 2ab = 2 \cdot 3 \cdot \sqrt{5}i = 6\sqrt{5}i \).
- Calculate \( b^2 = (\sqrt{5}i)^2 = 5i^2 = 5(-1) = -5 \).

Putting it all together:
\[ (3 + \sqrt{5}i)^2 = 9 + 6\sqrt{5}i - 5 = 4 + 6\sqrt{5}i \]

### Step 2: Find the multiplicative inverse
The multiplicative inverse of a complex number \( z \) is given by:
\[ z^{-1} = \frac{1}{z} \]
So we need to find:
\[ \frac{1}{4 + 6\sqrt{5}i} \]

### Step 3: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{4 + 6\sqrt{5}i} \cdot \frac{4 - 6\sqrt{5}i}{4 - 6\sqrt{5}i} = \frac{4 - 6\sqrt{5}i}{(4 + 6\sqrt{5}i)(4 - 6\sqrt{5}i)} \]

### Step 4: Calculate the denominator
Using the difference of squares:
\[ (4 + 6\sqrt{5}i)(4 - 6\sqrt{5}i) = 4^2 - (6\sqrt{5}i)^2 = 16 - 36 \cdot 5(-1) = 16 + 180 = 196 \]

### Step 5: Write the result
Now we can write the multiplicative inverse:
\[ \frac{4 - 6\sqrt{5}i}{196} \]
This simplifies to:
\[ \frac{4}{196} - \frac{6\sqrt{5}i}{196} = \frac{1}{49} - \frac{3\sqrt{5}i}{98} \]

### Final Answer
Thus, the multiplicative inverse of \( (3 + \sqrt{5}i)^2 \) is:
\[ \frac{1}{49} - \frac{3\sqrt{5}}{98}i \]