Find the multiplicative inverse of `3+4i`.

Answer

To find the multiplicative inverse of the complex number \(3 + 4i\), we follow these steps:

### Step 1: Understand the Concept of Multiplicative Inverse
The multiplicative inverse of a complex number \(z\) is another complex number \(w\) such that \(z \cdot w = 1\). For our case, we need to find \(w\) such that:
\[ (3 + 4i) \cdot w = 1 \]
This means we need to find \(w = \frac{1}{3 + 4i}\).

### Step 2: Rationalize the Denominator
To express \(w\) in the form \(x + yi\), we rationalize the denominator. We multiply the numerator and the denominator by the conjugate of the denominator:
\[ w = \frac{1}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i} \]
This gives us:
\[ w = \frac{3 - 4i}{(3 + 4i)(3 - 4i)} \]

### Step 3: Calculate the Denominator
Now, we calculate the denominator:
\[ (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \]

### Step 4: Substitute Back into the Expression
Now we substitute back into our expression for \(w\):
\[ w = \frac{3 - 4i}{25} \]
This can be separated into real and imaginary parts:
\[ w = \frac{3}{25} - \frac{4}{25}i \]

### Step 5: Write the Final Answer
Thus, the multiplicative inverse of \(3 + 4i\) is:
\[ \frac{3}{25} - \frac{4}{25}i \]

### Summary of Steps
1. Recognize that the multiplicative inverse is \(w = \frac{1}{3 + 4i}\).
2. Rationalize the denominator by multiplying by the conjugate.
3. Calculate the denominator using the formula \(a^2 + b^2\).
4. Substitute back to find \(w\) in the form \(x + yi\).
5. Present the final answer.