Find the multiplicative inverse of each of the following complex numbers when it exists.
`(3+4i)/(4-5i)`

Answer

To find the multiplicative inverse of the complex number \(\frac{3 + 4i}{4 - 5i}\), we will follow these steps:

### Step 1: Define the Complex Number
Let \( z = \frac{3 + 4i}{4 - 5i} \).

### Step 2: Write the Multiplicative Inverse
The multiplicative inverse of \( z \) is given by \( z^{-1} = \frac{1}{z} \). Thus, we can express this as:
\[ z^{-1} = \frac{1}{\frac{3 + 4i}{4 - 5i}} = \frac{4 - 5i}{3 + 4i} \]

### Step 3: Multiply by the Conjugate
To simplify \( z^{-1} \), we multiply the numerator and denominator by the conjugate of the denominator \( 3 - 4i \):
\[ z^{-1} = \frac{(4 - 5i)(3 - 4i)}{(3 + 4i)(3 - 4i)} \]

### Step 4: Calculate the Denominator
The denominator can be calculated using the formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \]

### Step 5: Calculate the Numerator
Now, we calculate the numerator:
\[ (4 - 5i)(3 - 4i) = 4 \cdot 3 + 4 \cdot (-4i) - 5i \cdot 3 - 5i \cdot (-4i) \]
\[ = 12 - 16i - 15i + 20 = 12 - 31i + 20 = 32 - 31i \]

### Step 6: Combine the Results
Now, we can combine the results from the numerator and denominator:
\[ z^{-1} = \frac{32 - 31i}{25} \]

### Step 7: Write in Standard Form
This can be expressed in standard form:
\[ z^{-1} = \frac{32}{25} - \frac{31}{25}i \]

### Final Answer
Thus, the multiplicative inverse of the complex number \(\frac{3 + 4i}{4 - 5i}\) is:
\[ \boxed{\frac{32}{25} - \frac{31}{25}i} \]