`A = (2+ 1) (22 + 1) (2 + 1)..... (2^(2048) + 1)` The value of `(A + 1)^(1//2048)` is

Answer

To solve the problem, we need to evaluate the expression \( A = (2^1 + 1)(2^2 + 1)(2^3 + 1) \cdots (2^{2048} + 1) \) and then find the value of \( (A + 1)^{\frac{1}{2048}} \).

### Step-by-Step Solution:

1. **Understanding the Expression for A**:
We have:
\[ A = (2^1 + 1)(2^2 + 1)(2^3 + 1) \cdots (2^{2048} + 1) \]

2. **Using the Formula**:
We can use the identity:
\[ (a - b)(a + b) = a^2 - b^2 \]
Here, we can rewrite \( 2^n + 1 \) in terms of \( 2^{n+1} - 1 \) and \( 2^{n+1} + 1 \).

3. **Rearranging A**:
Notice that:
\[ A = \frac{(2^{2048} - 1)(2^{2048} + 1)}{2 - 1} \]
This simplifies to:
\[ A = 2^{2048} - 1 \]

4. **Finding A + 1**:
Now, we can find \( A + 1 \):
\[ A + 1 = (2^{2048} - 1) + 1 = 2^{2048} \]

5. **Calculating \( (A + 1)^{\frac{1}{2048}} \)**:
We need to evaluate:
\[ (A + 1)^{\frac{1}{2048}} = (2^{2048})^{\frac{1}{2048}} = 2^{\frac{2048}{2048}} = 2^1 = 2 \]

6. **Final Answer**:
Therefore, the final value is:
\[ \boxed{2} \]