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 \( \bar{(z^{-1})} \) where \( z = 3 + i + 9i^2 - 6i^3 \).

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

First, we need to simplify \( z \) using the values of \( i^2 \) and \( i^3 \):
- We know that \( i^2 = -1 \) and \( i^3 = -i \).

Substituting these values into the equation:
\[ z = 3 + i + 9(-1) - 6(-i) \]
\[ = 3 + i - 9 + 6i \]
\[ = (3 - 9) + (1 + 6)i \]
\[ = -6 + 7i \]

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

Next, we need to find \( z^{-1} \), which is \( \frac{1}{z} \):
\[ z^{-1} = \frac{1}{-6 + 7i} \]

To compute this, we multiply the numerator and denominator by the conjugate of the denominator:
\[ z^{-1} = \frac{1 \cdot (-6 - 7i)}{(-6 + 7i)(-6 - 7i)} \]
Calculating the denominator:
\[ (-6 + 7i)(-6 - 7i) = (-6)^2 - (7i)^2 = 36 - 49(-1) = 36 + 49 = 85 \]
So, we have:
\[ z^{-1} = \frac{-6 - 7i}{85} \]
\[ = -\frac{6}{85} - \frac{7}{85}i \]

### Step 3: Find \( \bar{(z^{-1})} \)

Now we need to find the conjugate of \( z^{-1} \):
\[ \bar{(z^{-1})} = \bar{\left(-\frac{6}{85} - \frac{7}{85}i\right)} = -\frac{6}{85} + \frac{7}{85}i \]

### Final Answer

Thus, the final result is:
\[ \bar{(z^{-1})} = -\frac{6}{85} + \frac{7}{85}i \]

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