`[{(-(1)/(2))^(2)}^(-2)]^(-1)` is equal to :

Options

A. a) `(1)/(16)`
B. b) `16`
C. c) `-(1)/(16)`
D. d) `-16`

Answer

To solve the expression \([ \{ (-\frac{1}{2})^2 \} ^{-2} ]^{-1}\), we will follow the order of operations and the laws of exponents step by step.

### Step 1: Simplify the innermost expression
First, we simplify \((- \frac{1}{2})^2\):
\[ (-\frac{1}{2})^2 = \frac{1}{4} \]

### Step 2: Apply the next exponent
Now we substitute this back into the expression:
\[ \{ \frac{1}{4} \} ^{-2} \]
Using the property of exponents, \(a^{-n} = \frac{1}{a^n}\), we can rewrite this as:
\[ \frac{1}{(\frac{1}{4})^2} \]

### Step 3: Calculate \((\frac{1}{4})^2\)
Now we calculate \((\frac{1}{4})^2\):
\[ (\frac{1}{4})^2 = \frac{1}{16} \]

### Step 4: Substitute back into the expression
Now we substitute this back into our expression:
\[ \frac{1}{\frac{1}{16}} = 16 \]

### Step 5: Apply the final exponent
Now we take the final exponent:
\[ 16^{-1} = \frac{1}{16} \]

### Final Answer
Thus, the expression \([ \{ (-\frac{1}{2})^2 \} ^{-2} ]^{-1}\) simplifies to:
\[ \frac{1}{16} \]

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