The value of ` (sqrt(3+2sqrt2)+sqrt(3-2sqrt2))^(2^(9))` is ________.

Explanation

To solve the expression \( ( \sqrt{3 + 2\sqrt{2}} + \sqrt{3 - 2\sqrt{2}} )^{2^9} \), we will simplify the terms inside the parentheses first.

### Step 1: Simplify \( \sqrt{3 + 2\sqrt{2}} \)

We can express \( 3 + 2\sqrt{2} \) in a different form. Notice that:
\[ 3 + 2\sqrt{2} = (\sqrt{2} + 1)^2 \]
To verify:
\[ (\sqrt{2} + 1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \]
Thus,
\[ \sqrt{3 + 2\sqrt{2}} = \sqrt{(\sqrt{2} + 1)^2} = \sqrt{2} + 1 \]

### Step 2: Simplify \( \sqrt{3 - 2\sqrt{2}} \)

Similarly, we can express \( 3 - 2\sqrt{2} \):
\[ 3 - 2\sqrt{2} = (\sqrt{2} - 1)^2 \]
To verify:
\[ (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]
Thus,
\[ \sqrt{3 - 2\sqrt{2}} = \sqrt{(\sqrt{2} - 1)^2} = \sqrt{2} - 1 \]

### Step 3: Combine the results

Now we can combine the results:
\[ \sqrt{3 + 2\sqrt{2}} + \sqrt{3 - 2\sqrt{2}} = (\sqrt{2} + 1) + (\sqrt{2} - 1) = 2\sqrt{2} \]

### Step 4: Substitute back into the original expression

Now substitute back into the original expression:
\[ ( \sqrt{3 + 2\sqrt{2}} + \sqrt{3 - 2\sqrt{2}} )^{2^9} = (2\sqrt{2})^{2^9} \]

### Step 5: Simplify \( (2\sqrt{2})^{2^9} \)

We can rewrite \( 2\sqrt{2} \) as \( 2^{1} \cdot 2^{1/2} = 2^{3/2} \):
\[ (2\sqrt{2})^{2^9} = (2^{3/2})^{2^9} = 2^{(3/2) \cdot 2^9} = 2^{3 \cdot 2^8} = 2^{3 \cdot 256} = 2^{768} \]

### Final Answer

Thus, the value of \( ( \sqrt{3 + 2\sqrt{2}} + \sqrt{3 - 2\sqrt{2}} )^{2^9} \) is:
\[ \boxed{2^{768}} \]