Simplify `[{(256)^(-(1)/(2))}^(-(1)/(4))]^(2)`

Options

A. 3
B. 4
C. 5
D. 6

Answer

To simplify the expression \(\left[ (256)^{-\frac{1}{2}} \right]^{-\frac{1}{4}}^2\), we will follow these steps:

### Step 1: Rewrite 256 as a power of 2
First, we express 256 in terms of base 2:
\[ 256 = 2^8 \]
So we can rewrite the expression as:
\[ \left[ (2^8)^{-\frac{1}{2}} \right]^{-\frac{1}{4}}^2 \]

### Step 2: Apply the power of a power property
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we simplify:
\[ (2^8)^{-\frac{1}{2}} = 2^{8 \cdot -\frac{1}{2}} = 2^{-4} \]
Now our expression becomes:
\[ \left[ 2^{-4} \right]^{-\frac{1}{4}}^2 \]

### Step 3: Simplify the exponent
Again applying the power of a power property:
\[ \left[ 2^{-4} \right]^{-\frac{1}{4}} = 2^{-4 \cdot -\frac{1}{4}} = 2^{1} \]
Now we have:
\[ (2^1)^2 \]

### Step 4: Final simplification
Now we apply the power of a power property one last time:
\[ (2^1)^2 = 2^{1 \cdot 2} = 2^2 \]
Calculating \(2^2\):
\[ 2^2 = 4 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{4} \]