Express the following in the form a+bi
`(1-i)^(4)`

Answer

To express \((1 - i)^{4}\) in the form \(a + bi\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1 - i)^{4} \]
We can rewrite this as:
\[ ((1 - i)^{2})^{2} \]

### Step 2: Calculate \((1 - i)^{2}\)
Now, we need to calculate \((1 - i)^{2}\):
\[ (1 - i)^{2} = 1^{2} - 2 \cdot 1 \cdot i + i^{2} \]
Calculating each term:
- \(1^{2} = 1\)
- \(i^{2} = -1\)
- \(-2 \cdot 1 \cdot i = -2i\)

Putting it all together:
\[ (1 - i)^{2} = 1 - 2i - 1 = -2i \]

### Step 3: Calculate \((-2i)^{2}\)
Now, we substitute back into our expression:
\[ ((1 - i)^{2})^{2} = (-2i)^{2} \]
Calculating \((-2i)^{2}\):
\[ (-2i)^{2} = (-2)^{2} \cdot i^{2} = 4 \cdot (-1) = -4 \]

### Step 4: Write in the form \(a + bi\)
Now we have:
\[ (1 - i)^{4} = -4 \]
To express this in the form \(a + bi\), we can write:
\[ -4 + 0i \]
Thus, \(a = -4\) and \(b = 0\).

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

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