Value of `(2+1) (2^(2) +1) (2^(4) +1) (2^(8) +1) (2^(16) +1) (2^(32) +1) (2^(64) +1)` is

Options

A. `2^(256)-1`
B. `2^(256)+1`
C. `2^(128)-1`
D. `2^(128)+1`

Answer

To find the value of the expression \((2+1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1)(2^{64} + 1)\), we can use the formula for the difference of squares.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
\[ (2 + 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1)(2^{64} + 1) \]

2. **Recognize the Pattern**:
Notice that we can express this product in terms of a difference of squares:
\[ (2^1 - 1)(2^1 + 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1)(2^{64} + 1) \]
Here, we added \( (2 - 1) \) to the product.

3. **Apply the Difference of Squares**:
Using the identity \( a^2 - b^2 = (a - b)(a + b) \), we can pair the terms:
\[ (2^1 - 1)(2^1 + 1) = 2^2 - 1^2 = 4 - 1 = 3 \]
Now, we can express the remaining terms:
\[ (2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1)(2^{64} + 1) \]

4. **Continue Applying the Difference of Squares**:
Continuing this process:
- For \( (2^2 - 1)(2^2 + 1) = 2^4 - 1^2 = 16 - 1 = 15 \)
- For \( (2^4 - 1)(2^4 + 1) = 2^8 - 1^2 = 256 - 1 = 255 \)
- For \( (2^8 - 1)(2^8 + 1) = 2^{16} - 1^2 = 65536 - 1 = 65535 \)
- For \( (2^{16} - 1)(2^{16} + 1) = 2^{32} - 1^2 = 4294967296 - 1 = 4294967295 \)
- For \( (2^{32} - 1)(2^{32} + 1) = 2^{64} - 1^2 = 18446744073709551616 - 1 = 18446744073709551615 \)

5. **Final Calculation**:
The final result can be expressed as:
\[ 2^{128} - 1 \]
Therefore, the value of the original expression is:
\[ 2^{128} - 1 \]

### Final Answer:
\[ \text{Value} = 2^{128} - 1 \]