Perform the indicated operation and give your answer in the form `x+yi`, where x and y are real numbers and `i= sqrt(-1)`
`(3+4i)^(-1)`

Answer

To solve the problem \((3 + 4i)^{-1}\) and express the answer in the form \(x + yi\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (3 + 4i)^{-1} = \frac{1}{3 + 4i} \]

### Step 2: Multiply by the conjugate
To eliminate the imaginary part from the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(3 - 4i\):
\[ \frac{1}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i} = \frac{3 - 4i}{(3 + 4i)(3 - 4i)} \]

### Step 3: Calculate the denominator
Now we need to calculate the denominator \((3 + 4i)(3 - 4i)\). This can be simplified using the difference of squares formula:
\[ (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \]

### Step 4: Substitute back into the expression
Now we can substitute back into our expression:
\[ \frac{3 - 4i}{25} \]

### Step 5: Separate into real and imaginary parts
This can be separated into real and imaginary parts:
\[ \frac{3}{25} - \frac{4}{25}i \]

### Final Answer
Thus, the expression in the form \(x + yi\) is:
\[ \frac{3}{25} - \frac{4}{25}i \]