Express ` (1 -i)^(4)` in the form of a +ib.

Answer

To express \( (1 - i)^4 \) in the form of \( a + ib \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1 - i)^4 \]

### Step 2: Use the binomial theorem
We can apply the binomial theorem to expand \( (1 - i)^4 \):
\[ (1 - i)^4 = \sum_{k=0}^{4} \binom{4}{k} (1)^{4-k} (-i)^k \]
This expands to:
\[ = \binom{4}{0} (1)^4 (-i)^0 + \binom{4}{1} (1)^3 (-i)^1 + \binom{4}{2} (1)^2 (-i)^2 + \binom{4}{3} (1)^1 (-i)^3 + \binom{4}{4} (1)^0 (-i)^4 \]

### Step 3: Calculate each term
Calculating each term:
- For \( k=0 \): \( \binom{4}{0} (1)^4 (-i)^0 = 1 \)
- For \( k=1 \): \( \binom{4}{1} (1)^3 (-i)^1 = -4i \)
- For \( k=2 \): \( \binom{4}{2} (1)^2 (-i)^2 = 6(-1) = -6 \)
- For \( k=3 \): \( \binom{4}{3} (1)^1 (-i)^3 = -4(-i) = 4i \)
- For \( k=4 \): \( \binom{4}{4} (1)^0 (-i)^4 = 1 \)

### Step 4: Combine the terms
Now, we combine all the terms:
\[ 1 - 4i - 6 + 4i + 1 \]
The imaginary parts \( -4i + 4i \) cancel each other out, and we are left with:
\[ (1 - 6 + 1) = -4 \]

### Step 5: Write in the form \( a + ib \)
Thus, we can express the result as:
\[ -4 + 0i \]

### Final Answer
The expression \( (1 - i)^4 \) in the form \( a + ib \) is:
\[ -4 + 0i \]