Find the reciprocal of `3-4i`.

Options

A. `(3)/(25)+(4)/(25)i`.
B. `(3)/(25)-(4)/(25)i`.
C. `(1)/(25)+(4)/(25)i`.
D. `(4)/(25)+(3)/(25)i`.

Answer

To find the reciprocal of the complex number \(3 - 4i\), we can follow these steps:

### Step 1: Write the reciprocal
The reciprocal of a complex number \(z = a + bi\) is given by \(\frac{1}{z}\). Therefore, the reciprocal of \(3 - 4i\) is:
\[ \frac{1}{3 - 4i} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 - 4i\) is \(3 + 4i\). Thus, we have:
\[ \frac{1}{3 - 4i} \cdot \frac{3 + 4i}{3 + 4i} = \frac{3 + 4i}{(3 - 4i)(3 + 4i)} \]

### 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: Write the result
Now substituting back into our expression, we get:
\[ \frac{3 + 4i}{25} \]

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

### Final Result
Thus, the reciprocal of \(3 - 4i\) is:
\[ \frac{3}{25} + \frac{4}{25}i \]

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