Find the multiplicative inverse of `7+24i`

Answer

To find the multiplicative inverse of the complex number \( 7 + 24i \), we will follow these steps:

### Step 1: Write the expression for the multiplicative inverse
The multiplicative inverse of a complex number \( z \) is given by \( z^{-1} = \frac{1}{z} \). In this case, we have:
\[ z = 7 + 24i \]
Thus, the multiplicative inverse is:
\[ z^{-1} = \frac{1}{7 + 24i} \]

### Step 2: Rationalize the denominator
To simplify \( \frac{1}{7 + 24i} \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 7 - 24i \):
\[ z^{-1} = \frac{1 \cdot (7 - 24i)}{(7 + 24i)(7 - 24i)} = \frac{7 - 24i}{(7 + 24i)(7 - 24i)} \]

### Step 3: Calculate the denominator
Now, we need to compute the denominator \( (7 + 24i)(7 - 24i) \):
\[ (7 + 24i)(7 - 24i) = 7^2 - (24i)^2 = 49 - 576(-1) = 49 + 576 = 625 \]

### Step 4: Write the final expression
Now, substituting back into our expression for the multiplicative inverse, we have:
\[ z^{-1} = \frac{7 - 24i}{625} \]

### Step 5: Separate into real and imaginary parts
This can be expressed as:
\[ z^{-1} = \frac{7}{625} - \frac{24}{625}i \]

### Final Result
Thus, the multiplicative inverse of \( 7 + 24i \) is:
\[ \frac{7}{625} - \frac{24}{625}i \]

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