Find the multiplicative inverse of the following
`sqrt5 + 3i`

Answer

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

### Step 1: Write the expression for the multiplicative inverse
The multiplicative inverse of a complex number \( z \) is given by \( z^{-1} = \frac{1}{z} \). Here, our complex number \( z = \sqrt{5} + 3i \).

\[ z^{-1} = \frac{1}{\sqrt{5} + 3i} \]

### Step 2: Rationalize the denominator
To simplify \( \frac{1}{\sqrt{5} + 3i} \), we will multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{5} - 3i \).

\[ z^{-1} = \frac{1 \cdot (\sqrt{5} - 3i)}{(\sqrt{5} + 3i)(\sqrt{5} - 3i)} \]

### Step 3: Expand the denominator using the difference of squares
The denominator can be simplified using the formula \( (a + b)(a - b) = a^2 - b^2 \).

\[ (\sqrt{5})^2 - (3i)^2 = 5 - 9(-1) = 5 + 9 = 14 \]

### Step 4: Write the simplified expression
Now substituting back into our expression for the multiplicative inverse:

\[ z^{-1} = \frac{\sqrt{5} - 3i}{14} \]

### Step 5: Final result
Thus, the multiplicative inverse of \( \sqrt{5} + 3i \) is:

\[ \frac{\sqrt{5}}{14} - \frac{3i}{14} \]

### Summary
The multiplicative inverse of \( \sqrt{5} + 3i \) is \( \frac{\sqrt{5}}{14} - \frac{3i}{14} \).

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