The value of ` (1+i) (1-i^(2)) (1+i^(4))(1-i^(5))` is

Options

A. 2i
B. 8
C. `-8`
D. 8i

Answer

To solve the expression \( (1+i)(1-i^2)(1+i^4)(1-i^5) \), we will simplify each term step by step.

### Step 1: Simplify \( 1 - i^2 \)
We know that \( i^2 = -1 \). Therefore:
\[ 1 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 2: Simplify \( 1 + i^4 \)
We know that \( i^4 = 1 \). Therefore:
\[ 1 + i^4 = 1 + 1 = 2 \]

### Step 3: Simplify \( 1 - i^5 \)
We can express \( i^5 \) as \( i^4 \cdot i = 1 \cdot i = i \). Therefore:
\[ 1 - i^5 = 1 - i \]

### Step 4: Combine the results
Now we can substitute back into the original expression:
\[ (1+i)(2)(2)(1-i) \]
This simplifies to:
\[ 4(1+i)(1-i) \]

### Step 5: Simplify \( (1+i)(1-i) \)
Using the difference of squares:
\[ (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 6: Final calculation
Now substituting back:
\[ 4 \cdot 2 = 8 \]

Thus, the value of the expression \( (1+i)(1-i^2)(1+i^4)(1-i^5) \) is \( \boxed{8} \).

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