What is the simplified value of `(2+1)(2^(2)+1)(2^(4)+1)(2^(8)+1)` ?

Options

A. A) `2^(8)-1`
B. B) `2^(16)-1`
C. C) `2^(32)-1`
D. D) `2^(64)-1`

Answer

To simplify the expression \((2+1)(2^2+1)(2^4+1)(2^8+1)\), we can follow these steps:

### Step 1: Evaluate Each Term
First, we evaluate each term in the expression:
- \(2 + 1 = 3\)
- \(2^2 + 1 = 4 + 1 = 5\)
- \(2^4 + 1 = 16 + 1 = 17\)
- \(2^8 + 1 = 256 + 1 = 257\)

### Step 2: Multiply the Terms Together
Now we multiply these results together:
\[ 3 \times 5 \times 17 \times 257 \]

### Step 3: Simplify the Multiplication
We can calculate the multiplication step by step:
1. \(3 \times 5 = 15\)
2. \(15 \times 17 = 255\)
3. \(255 \times 257\)

### Step 4: Calculate \(255 \times 257\)
To calculate \(255 \times 257\), we can use the difference of squares:
\[ 255 \times 257 = (256 - 1)(256 + 1) = 256^2 - 1^2 = 65536 - 1 = 65535 \]

### Step 5: Final Result
Thus, the simplified value of \((2+1)(2^2+1)(2^4+1)(2^8+1)\) is \(65535\).

### Step 6: Express in Power of 2
Notice that \(65535\) can be expressed as \(2^{16} - 1\) since:
\[ 65535 = 2^{16} - 1 \]

### Final Answer
The simplified value of \((2+1)(2^2+1)(2^4+1)(2^8+1)\) is:
\[ \boxed{2^{16} - 1} \]