` (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 can follow these steps:

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

### Step 2: Use the property of exponents
We can rewrite \( (1+i)^4 \) and \( (1-i)^4 \) as:
\[ ((1+i)^2)^2 + ((1-i)^2)^2 \]

### Step 3: Calculate \( (1+i)^2 \) and \( (1-i)^2 \)
Now, we calculate \( (1+i)^2 \) and \( (1-i)^2 \):
\[ (1+i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i - 1 = 2i \]
\[ (1-i)^2 = 1^2 - 2 \cdot 1 \cdot i + i^2 = 1 - 2i - 1 = -2i \]

### Step 4: Substitute back into the expression
Now substitute these results back into the expression:
\[ (2i)^2 + (-2i)^2 \]

### Step 5: Calculate \( (2i)^2 \) and \( (-2i)^2 \)
Calculating these squares gives:
\[ (2i)^2 = 4i^2 = 4(-1) = -4 \]
\[ (-2i)^2 = 4i^2 = 4(-1) = -4 \]

### Step 6: Combine the results
Now we can combine these results:
\[ -4 + (-4) = -8 \]

### Final Answer
Thus, the value of the expression \( (1+i)^4 + (1-i)^4 \) is:
\[ \boxed{-8} \]