Find the multiplicative inverse of the following
`(3,4)`

Answer

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

### Step 1: Represent the complex number
The complex number can be represented as:
\[ z = 3 + 4i \]

### Step 2: Write the formula for the multiplicative inverse
The multiplicative inverse of a complex number \( z \) is given by:
\[ z^{-1} = \frac{1}{z} \]

### Step 3: Substitute the complex number into the formula
Substituting \( z \) into the formula, we have:
\[ z^{-1} = \frac{1}{3 + 4i} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator:
\[ z^{-1} = \frac{1}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i} = \frac{3 - 4i}{(3 + 4i)(3 - 4i)} \]

### Step 5: Calculate the denominator
Now, calculate the denominator using the formula \( (a + bi)(a - bi) = a^2 + b^2 \):
\[ (3 + 4i)(3 - 4i) = 3^2 + (4)^2 = 9 + 16 = 25 \]

### Step 6: Write the result
Now, substituting back into our expression for the multiplicative inverse:
\[ z^{-1} = \frac{3 - 4i}{25} \]

### Step 7: Simplify the expression
This can be expressed as:
\[ z^{-1} = \frac{3}{25} - \frac{4}{25}i \]

Thus, the multiplicative inverse of the complex number \( (3, 4) \) is:
\[ \frac{3}{25} - \frac{4}{25}i \]

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