If z is any non-zero complex number, prove that the multiplicative inverse of z is `(overline(z))/(|z|^(2))`.
Hence, express `(4-sqrt(-9))^(-1)` in the form x+iy, where x,y `inR`

Answer

To prove that the multiplicative inverse of a non-zero complex number \( z \) is given by \( \frac{\overline{z}}{|z|^2} \), and to express \( (4 - \sqrt{-9})^{-1} \) in the form \( x + iy \), we will follow these steps:

### Step 1: Define the Complex Number
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit.

### Step 2: Find the Multiplicative Inverse
The multiplicative inverse of \( z \) is defined as \( z^{-1} \) such that:
\[ z \cdot z^{-1} = 1 \]
We can express \( z^{-1} \) as:
\[ z^{-1} = \frac{1}{z} = \frac{1}{x + iy} \]

### Step 3: Rationalize the Denominator
To simplify \( \frac{1}{x + iy} \), we multiply the numerator and denominator by the conjugate of the denominator:
\[ z^{-1} = \frac{1 \cdot (x - iy)}{(x + iy)(x - iy)} = \frac{x - iy}{x^2 + y^2} \]
Here, \( (x + iy)(x - iy) = x^2 + y^2 \) because \( i^2 = -1 \).

### Step 4: Express in Terms of Modulus
The modulus \( |z| \) of the complex number \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
Thus, \( |z|^2 = x^2 + y^2 \).

### Step 5: Final Form of the Inverse
Therefore, we can rewrite the multiplicative inverse as:
\[ z^{-1} = \frac{x - iy}{|z|^2} = \frac{\overline{z}}{|z|^2} \]
This proves that the multiplicative inverse of \( z \) is indeed \( \frac{\overline{z}}{|z|^2} \).

### Step 6: Find the Inverse of \( (4 - \sqrt{-9}) \)
Now we need to express \( (4 - \sqrt{-9})^{-1} \):
First, we simplify \( \sqrt{-9} \):
\[ \sqrt{-9} = 3i \]
Thus, we have:
\[ z = 4 - 3i \]

### Step 7: Find the Conjugate and Modulus
The conjugate \( \overline{z} \) is:
\[ \overline{z} = 4 + 3i \]
Next, we calculate \( |z|^2 \):
\[ |z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25 \]

### Step 8: Calculate the Inverse
Now we can find the multiplicative inverse:
\[ z^{-1} = \frac{\overline{z}}{|z|^2} = \frac{4 + 3i}{25} \]
This can be expressed as:
\[ z^{-1} = \frac{4}{25} + \frac{3}{25}i \]

### Conclusion
Thus, the multiplicative inverse of \( (4 - \sqrt{-9}) \) in the form \( x + iy \) is:
\[ \frac{4}{25} + \frac{3}{25}i \]