The multiplicative inverse of 3+4i is

Options

A. (a) 3-4i
B. (b) `(3+4i)/(25)`
C. (c) `(3-4i)/(25)`
D. (d) `-3+4i`

Answer

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

### Step 1: Define the Complex Number
Let \( z = 3 + 4i \).

### Step 2: Write the Formula for the Multiplicative Inverse
The multiplicative inverse of a complex number \( z \) is given by:
\[ \frac{1}{z} \]
Thus, we need to compute:
\[ \frac{1}{3 + 4i} \]

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

### Step 4: Simplify the Denominator
Now we simplify the denominator using the formula for the difference of squares:
\[ (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - 16i^2 \]
Since \(i^2 = -1\), we have:
\[ 9 - 16(-1) = 9 + 16 = 25 \]

### Step 5: Write the Result
Now we can write the expression for the multiplicative inverse:
\[ \frac{3 - 4i}{25} \]

### Step 6: Final Answer
Thus, the multiplicative inverse of \(3 + 4i\) is:
\[ \frac{3 - 4i}{25} \]

### Conclusion
The correct answer is \(\frac{3 - 4i}{25}\).

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