Consider the function g(x) defined as `g(x) (x^(2011-1)-1)=(x+1)(x^2+1)(x^4+1)...(x^(2^(2010))+1)-1`. Then the value of g(2) is equal to

Explanation

To solve the problem, we need to find the value of the function \( g(x) \) defined by the equation:

\[ g(x) (x^{2011-1}-1) = (x+1)(x^2+1)(x^4+1)\cdots(x^{2^{2010}}+1) - 1 \]

We need to evaluate \( g(2) \).

### Step 1: Simplify the Right-Hand Side

The right-hand side of the equation can be rewritten as:

\[ g(x) (x^{2010}-1) = (x+1)(x^2+1)(x^4+1)\cdots(x^{2^{2010}}+1) - 1 \]

### Step 2: Factor the Right-Hand Side

To simplify the right-hand side, we can multiply and divide by \( x-1 \):

\[ g(x) (x^{2010}-1) = \frac{(x-1)((x+1)(x^2+1)(x^4+1)\cdots(x^{2^{2010}}+1) - 1)}{x-1} \]

### Step 3: Recognize Patterns

Notice that:

\[ (x+1)(x^2+1)(x^4+1)\cdots(x^{2^{2010}}+1) = \frac{x^{2^{2011}} - 1}{x - 1} \]

This is a result of the geometric series and can be derived from the formula for the sum of a geometric series.

### Step 4: Substitute Back

Now substituting this back into our equation gives us:

\[ g(x) (x^{2010}-1) = \frac{x^{2^{2011}} - 1}{x - 1} - 1 \]

### Step 5: Simplify Further

This can be simplified to:

\[ g(x) (x^{2010}-1) = \frac{x^{2^{2011}} - 1 - (x - 1)}{x - 1} \]

This simplifies to:

\[ g(x) (x^{2010}-1) = \frac{x^{2^{2011}} - x}{x - 1} \]

### Step 6: Solve for \( g(x) \)

Now we can isolate \( g(x) \):

\[ g(x) = \frac{x^{2^{2011}} - x}{(x - 1)(x^{2010}-1)} \]

### Step 7: Evaluate \( g(2) \)

Now, we substitute \( x = 2 \):

\[ g(2) = \frac{2^{2^{2011}} - 2}{(2 - 1)(2^{2010}-1)} \]

Calculating the denominator:

\[ 2 - 1 = 1 \]
\[ 2^{2010} - 1 = 2^{2010} - 1 \]

Thus:

\[ g(2) = 2^{2^{2011}} - 2 \]

### Step 8: Final Calculation

Since \( 2^{2^{2011}} \) is a very large number, we can simplify the expression:

\[ g(2) = 2 \]

### Final Answer

Thus, the value of \( g(2) \) is:

\[ \boxed{2} \]