If `z=8-2i,z^(2)=`

Options

A. `60-32i`
B. `64+4i`
C. `64-4i`
D. `60`

Explanation

To find \( z^2 \) where \( z = 8 - 2i \), we will follow these steps:

### Step 1: Write the expression for \( z^2 \)
Given \( z = 8 - 2i \), we can express \( z^2 \) as:
\[ z^2 = (8 - 2i)^2 \]

**Hint:** Remember that squaring a binomial involves using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \).

### Step 2: Apply the binomial expansion
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
- Let \( a = 8 \) and \( b = 2i \).
\[ z^2 = 8^2 - 2 \cdot 8 \cdot (2i) + (2i)^2 \]

**Hint:** Calculate each term separately: \( a^2 \), \( -2ab \), and \( b^2 \).

### Step 3: Calculate each term
1. Calculate \( 8^2 \):
\[ 8^2 = 64 \]

2. Calculate \( -2 \cdot 8 \cdot (2i) \):
\[ -2 \cdot 8 \cdot (2i) = -32i \]

3. Calculate \( (2i)^2 \):
\[ (2i)^2 = 4i^2 = 4(-1) = -4 \]

**Hint:** Remember that \( i^2 = -1 \).

### Step 4: Combine the results
Now substitute the calculated values back into the expression:
\[ z^2 = 64 - 32i - 4 \]

### Step 5: Simplify the expression
Combine the real parts:
\[ z^2 = (64 - 4) - 32i = 60 - 32i \]

### Final Answer
Thus, the value of \( z^2 \) is:
\[ \boxed{60 - 32i} \]