`[{(-1/2)^(2)}^(-2)]^(-1)=?`

Options

A. `1/16`
B. 16
C. `-1/16`
D. -16

Answer

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

### Step 1: Simplify the innermost exponent
We start with the expression \((- \frac{1}{2})^2\).
\[ (-\frac{1}{2})^2 = \frac{1}{4} \]

### Step 2: Apply the next exponent
Now we take the result from Step 1 and raise it to the power of \(-2\):
\[ (\frac{1}{4})^{-2} \]
Using the property of exponents that states \(a^{-n} = \frac{1}{a^n}\), we can rewrite this as:
\[ \frac{1}{(\frac{1}{4})^2} \]
Calculating \((\frac{1}{4})^2\):
\[ (\frac{1}{4})^2 = \frac{1}{16} \]
Thus,
\[ \frac{1}{(\frac{1}{4})^2} = \frac{1}{\frac{1}{16}} = 16 \]

### Step 3: Apply the outermost exponent
Now we take the result from Step 2 and raise it to the power of \(-1\):
\[ 16^{-1} \]
Using the property of exponents again:
\[ 16^{-1} = \frac{1}{16} \]

### Final Result
Thus, the final result of the expression \([{{(-\frac{1}{2})}^{2}}^{-2}]^{-1}\) is:
\[ \frac{1}{16} \]