` (1+i)^(4)+(1-i)^( 4)` is equal to

Options

A. 8
B. -4
C. -8
D. None of these

Answer

To solve the expression \( (1+i)^{4} + (1-i)^{4} \), we will follow these steps:

### Step 1: Calculate \( (1+i)^{2} \)

Using the formula for the square of a binomial, \( (a+b)^{2} = a^{2} + 2ab + b^{2} \):

\[ (1+i)^{2} = 1^{2} + 2 \cdot 1 \cdot i + i^{2} \]
\[ = 1 + 2i + (-1) = 2i \]

### Step 2: Calculate \( (1+i)^{4} \)

Now, we will square the result from Step 1:

\[ (1+i)^{4} = (2i)^{2} = 4i^{2} \]
\[ = 4(-1) = -4 \]

### Step 3: Calculate \( (1-i)^{2} \)

Similarly, we calculate \( (1-i)^{2} \):

\[ (1-i)^{2} = 1^{2} - 2 \cdot 1 \cdot i + i^{2} \]
\[ = 1 - 2i + (-1) = -2i \]

### Step 4: Calculate \( (1-i)^{4} \)

Now, we will square the result from Step 3:

\[ (1-i)^{4} = (-2i)^{2} = 4i^{2} \]
\[ = 4(-1) = -4 \]

### Step 5: Combine the results

Now we can combine the results from Step 2 and Step 4:

\[ (1+i)^{4} + (1-i)^{4} = -4 + (-4) = -8 \]

### Final Answer

Thus, the value of \( (1+i)^{4} + (1-i)^{4} \) is:

\[ \boxed{-8} \]