if ` z = 3+i+9i^(2) -6i^(3) " then " (bar(z^(-1)))` is

Options

A. 2+i
B. ` - 3/79 + 4/79 i`
C. `1-i`
D. ` - 6/85 + 7/85 i`

Answer

To solve the problem, we need to find the conjugate of the inverse of the complex number \( z \) given by:

\[ z = 3 + i + 9i^2 - 6i^3 \]

### Step 1: Simplify \( z \)

First, we need to substitute the values of \( i^2 \) and \( i^3 \):
- \( i^2 = -1 \)
- \( i^3 = -i \)

Now, substituting these values into the equation for \( z \):

\[ z = 3 + i + 9(-1) - 6(-i) \]

This simplifies to:

\[ z = 3 + i - 9 + 6i \]

Combining like terms:

\[ z = (3 - 9) + (1 + 6)i = -6 + 7i \]

### Step 2: Find \( z^{-1} \)

The inverse of \( z \) is given by:

\[ z^{-1} = \frac{1}{z} = \frac{1}{-6 + 7i} \]

To simplify this, we multiply the numerator and denominator by the conjugate of the denominator:

\[ z^{-1} = \frac{1 \cdot (-6 - 7i)}{(-6 + 7i)(-6 - 7i)} \]

### Step 3: Simplify the denominator

Calculating the denominator:

\[ (-6 + 7i)(-6 - 7i) = (-6)^2 - (7i)^2 = 36 - 49(-1) = 36 + 49 = 85 \]

### Step 4: Write \( z^{-1} \)

Now substituting back, we have:

\[ z^{-1} = \frac{-6 - 7i}{85} \]

### Step 5: Find \( \overline{z^{-1}} \)

The conjugate of \( z^{-1} \) is obtained by changing the sign of the imaginary part:

\[ \overline{z^{-1}} = \frac{-6 + 7i}{85} \]

### Final Result

Thus, the value of \( \overline{z^{-1}} \) is:

\[ \overline{z^{-1}} = -\frac{6}{85} + \frac{7}{85}i \]