Find the multiplicative inverse of the product of complex numbers:
`3+4i,5-12i`.

Answer

To find the multiplicative inverse of the product of the complex numbers \(3 + 4i\) and \(5 - 12i\), we will follow these steps:

### Step 1: Calculate the product of the complex numbers
We start by multiplying the two complex numbers:
\[ (3 + 4i)(5 - 12i) \]

Using the distributive property (FOIL method):
\[ = 3 \cdot 5 + 3 \cdot (-12i) + 4i \cdot 5 + 4i \cdot (-12i) \]
\[ = 15 - 36i + 20i - 48i^2 \]

### Step 2: Simplify the expression
Recall that \(i^2 = -1\), so we can replace \(i^2\) in our expression:
\[ = 15 - 36i + 20i + 48 \]
Combine like terms:
\[ = (15 + 48) + (-36i + 20i) \]
\[ = 63 - 16i \]

### Step 3: Find the multiplicative inverse
The multiplicative inverse of a complex number \(z = a + bi\) is given by:
\[ \frac{1}{z} = \frac{1}{a + bi} \]
To find the multiplicative inverse of \(63 - 16i\), we need to rationalize the denominator:
\[ \frac{1}{63 - 16i} \cdot \frac{63 + 16i}{63 + 16i} = \frac{63 + 16i}{(63 - 16i)(63 + 16i)} \]

### Step 4: Calculate the denominator
Now we calculate the denominator:
\[ (63 - 16i)(63 + 16i) = 63^2 - (16i)^2 \]
\[ = 3969 - 256(-1) = 3969 + 256 = 4225 \]

### Step 5: Write the multiplicative inverse
Now we can write the multiplicative inverse:
\[ \frac{63 + 16i}{4225} \]
This can be expressed as:
\[ \frac{63}{4225} + \frac{16}{4225}i \]

### Final Answer
Thus, the multiplicative inverse of the product of the complex numbers \(3 + 4i\) and \(5 - 12i\) is:
\[ \frac{63}{4225} + \frac{16}{4225}i \]