Express the following in the form a + ib
`(1-i)^3 (1- i)`

Answer

To express \((1-i)^3(1-i)\) in the form \(a + ib\), we can follow these steps:

### Step 1: Simplify the expression
We start with the expression:
\[ (1-i)^3(1-i) \]
This can be rewritten as:
\[ (1-i)^4 \]

### Step 2: Expand \((1-i)^4\)
We can use the binomial theorem or simply expand it step by step. First, we find \((1-i)^2\):
\[ (1-i)^2 = 1^2 - 2(1)(i) + i^2 = 1 - 2i - 1 = -2i \]
Now we can square this result to find \((1-i)^4\):
\[ (1-i)^4 = (-2i)^2 = 4i^2 \]

### Step 3: Substitute \(i^2\)
We know that \(i^2 = -1\), so:
\[ 4i^2 = 4(-1) = -4 \]

### Step 4: Write in the form \(a + ib\)
We can express \(-4\) in the form \(a + ib\):
\[ -4 + 0i \]
Thus, \(a = -4\) and \(b = 0\).

### Final Result
The expression \((1-i)^3(1-i)\) in the form \(a + ib\) is:
\[ -4 + 0i \]