The expression `[(sqrt2)^(sqrt2)]^(sqrt2)` gives

Options

A. a natural number
B. a integer and not a natural number
C. a rational number but not an integer
D. a real number but not a rational number

Answer

To solve the expression \([( \sqrt{2} )^{\sqrt{2}}]^{\sqrt{2}}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ [(\sqrt{2})^{\sqrt{2}}]^{\sqrt{2}} \]

### Step 2: Apply the power of a power rule
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we can simplify the expression:
\[ (\sqrt{2})^{\sqrt{2} \cdot \sqrt{2}} \]

### Step 3: Simplify the exponent
Next, we simplify the exponent \(\sqrt{2} \cdot \sqrt{2}\):
\[ \sqrt{2} \cdot \sqrt{2} = 2 \]
Thus, the expression becomes:
\[ (\sqrt{2})^2 \]

### Step 4: Simplify \((\sqrt{2})^2\)
Now, we simplify \((\sqrt{2})^2\):
\[ (\sqrt{2})^2 = 2 \]

### Final Answer
Therefore, the expression \([( \sqrt{2} )^{\sqrt{2}}]^{\sqrt{2}}\) evaluates to:
\[ 2 \]