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

Options

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

Answer

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

### Step 1: Rewrite the expression
The expression can be rewritten as:
\[ (3)(2^2 + 1)(2^4 + 1) \]

### Step 2: Simplify \(2^2 + 1\)
Calculate \(2^2 + 1\):
\[ 2^2 + 1 = 4 + 1 = 5 \]
So now we have:
\[ 3 \cdot 5 \cdot (2^4 + 1) \]

### Step 3: Simplify \(2^4 + 1\)
Calculate \(2^4 + 1\):
\[ 2^4 + 1 = 16 + 1 = 17 \]
Now the expression becomes:
\[ 3 \cdot 5 \cdot 17 \]

### Step 4: Calculate \(3 \cdot 5\)
Multiply \(3\) and \(5\):
\[ 3 \cdot 5 = 15 \]
Now we have:
\[ 15 \cdot 17 \]

### Step 5: Calculate \(15 \cdot 17\)
Now we multiply \(15\) and \(17\):
\[ 15 \cdot 17 = 255 \]

### Step 6: Factorization Approach
Alternatively, we can use a factorization approach:
\[ (2 + 1)(2^2 + 1)(2^4 + 1) = \frac{2^8 - 1}{2 - 1} = 2^8 - 1 \]
This is derived from the identity:
\[ (2^n - 1) = (2 - 1)(2^{n-1} + 2^{n-2} + ... + 1) \]
Thus, we can conclude that:
\[ (2 + 1)(2^2 + 1)(2^4 + 1) = 2^8 - 1 \]

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