if ` z = 2 + i + 4i^(2) -6i^(3)` then verify that
(i) ` (bar(z^(2)) = (barz)^(2))`

Answer

To verify that \( \overline{z^2} = \overline{z}^2 \) for the complex number \( z = 2 + i + 4i^2 - 6i^3 \), we will follow these steps:

### Step 1: Simplify \( z \)
Given:
\[ z = 2 + i + 4i^2 - 6i^3 \]
We know:
- \( i^2 = -1 \)
- \( i^3 = -i \)

Substituting these values into \( z \):
\[ z = 2 + i + 4(-1) - 6(-i) \]
\[ = 2 + i - 4 + 6i \]
\[ = (2 - 4) + (1 + 6)i \]
\[ = -2 + 7i \]

### Step 2: Calculate \( z^2 \)
Now, we need to find \( z^2 \):
\[ z^2 = (-2 + 7i)(-2 + 7i) \]
Using the formula \( (a + b)(a + b) = a^2 + 2ab + b^2 \):
\[ = (-2)^2 + 2(-2)(7i) + (7i)^2 \]
\[ = 4 - 28i + 49i^2 \]
Since \( i^2 = -1 \):
\[ = 4 - 28i - 49 \]
\[ = -45 - 28i \]

### Step 3: Find \( \overline{z^2} \)
The complex conjugate of \( z^2 \) is:
\[ \overline{z^2} = \overline{-45 - 28i} = -45 + 28i \]

### Step 4: Find \( \overline{z} \)
Now, we find the complex conjugate of \( z \):
\[ \overline{z} = \overline{-2 + 7i} = -2 - 7i \]

### Step 5: Calculate \( \overline{z}^2 \)
Now, we need to calculate \( \overline{z}^2 \):
\[ \overline{z}^2 = (-2 - 7i)(-2 - 7i) \]
Using the same formula:
\[ = (-2)^2 + 2(-2)(-7i) + (-7i)^2 \]
\[ = 4 + 28i + 49i^2 \]
Substituting \( i^2 = -1 \):
\[ = 4 + 28i - 49 \]
\[ = -45 + 28i \]

### Step 6: Verify the equality
Now we compare \( \overline{z^2} \) and \( \overline{z}^2 \):
\[ \overline{z^2} = -45 + 28i \]
\[ \overline{z}^2 = -45 + 28i \]
Thus, we have:
\[ \overline{z^2} = \overline{z}^2 \]

### Conclusion
We have verified that \( \overline{z^2} = \overline{z}^2 \).

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