If the multiplicative inverse of a complex number is `(sqrt(5) + 6i)/(41)` then the complex number itself is :

Options

A. `sqrt(5) - 6i`
B. `sqrt(5) + 6i`
C. `6+ sqrt(7)i`
D. none of these

Answer

To find the complex number given that its multiplicative inverse is \((\sqrt{5} + 6i)/(41)\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Multiplicative Inverse**:
The multiplicative inverse of a complex number \( z \) is defined as \( z^{-1} = \frac{1}{z} \). Given that \( z^{-1} = \frac{\sqrt{5} + 6i}{41} \), we can express \( z \) in terms of \( z^{-1} \).

**Hint**: Remember that if \( z^{-1} = \frac{1}{z} \), then \( z = \frac{1}{z^{-1}} \).

2. **Finding the Complex Number**:
To find \( z \), we compute:
\[ z = \frac{1}{z^{-1}} = \frac{1}{\frac{\sqrt{5} + 6i}{41}} = \frac{41}{\sqrt{5} + 6i} \]

**Hint**: When dividing by a complex number, it helps to rationalize the denominator.

3. **Rationalizing the Denominator**:
We multiply the numerator and the denominator by the conjugate of the denominator:
\[ z = \frac{41}{\sqrt{5} + 6i} \cdot \frac{\sqrt{5} - 6i}{\sqrt{5} - 6i} = \frac{41(\sqrt{5} - 6i)}{(\sqrt{5} + 6i)(\sqrt{5} - 6i)} \]

**Hint**: The product of a complex number and its conjugate gives a real number.

4. **Calculating the Denominator**:
The denominator simplifies as follows:
\[ (\sqrt{5})^2 - (6i)^2 = 5 - 36(-1) = 5 + 36 = 41 \]

**Hint**: Remember that \( i^2 = -1 \).

5. **Final Simplification**:
Now substituting back into the expression for \( z \):
\[ z = \frac{41(\sqrt{5} - 6i)}{41} = \sqrt{5} - 6i \]

**Hint**: When you have a common factor in the numerator and denominator, you can cancel it out.

### Final Answer:
The complex number is:
\[ z = \sqrt{5} - 6i \]