If z `= (1-i)/(1+i)` then `z^(4)` equals

Options

A. 1
B. `-1`
C. 0
D. none of these

Answer

To solve the problem where \( z = \frac{1 - i}{1 + i} \) and we need to find \( z^4 \), we can follow these steps:

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

We start with the expression for \( z \):
\[ z = \frac{1 - i}{1 + i} \]

To simplify this, we can multiply the numerator and the denominator by the conjugate of the denominator:
\[ z = \frac{(1 - i)(1 - i)}{(1 + i)(1 - i)} \]

### Step 2: Calculate the denominator

Calculating the denominator:
\[ (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 3: Calculate the numerator

Now, calculating the numerator:
\[ (1 - i)(1 - i) = 1^2 - 2 \cdot 1 \cdot i + i^2 = 1 - 2i - 1 = -2i \]

### Step 4: Combine results

Now we can combine the results:
\[ z = \frac{-2i}{2} = -i \]

### Step 5: Calculate \( z^2 \)

Next, we find \( z^2 \):
\[ z^2 = (-i)^2 = (-1)^2 \cdot i^2 = 1 \cdot (-1) = -1 \]

### Step 6: Calculate \( z^4 \)

Now we can find \( z^4 \) by squaring \( z^2 \):
\[ z^4 = (z^2)^2 = (-1)^2 = 1 \]

### Final Answer

Thus, the value of \( z^4 \) is:
\[ \boxed{1} \]

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