`([(12)^(-2)]^(2))/([(12)^(2)]^(-2))=?`

Options

A. 12
B. 4.8
C. `12/144`
D. 1

Answer

To solve the expression \(\frac{(12)^{-2})^{2}}{(12)^{2}}^{-2}\), we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is \((12)^{-2})^{2}\). According to the power of a power property, we multiply the exponents:
\[ (12)^{-2})^{2} = 12^{-2 \cdot 2} = 12^{-4} \]

### Step 2: Simplify the Denominator
The denominator is \((12)^{2})^{-2}\). Again, using the power of a power property, we multiply the exponents:
\[ (12)^{2})^{-2} = 12^{2 \cdot -2} = 12^{-4} \]

### Step 3: Write the Fraction
Now we can rewrite the expression with the simplified numerator and denominator:
\[ \frac{12^{-4}}{12^{-4}} \]

### Step 4: Apply the Quotient Rule
Since the bases are the same, we can apply the quotient rule, which states that when dividing like bases, we subtract the exponents:
\[ 12^{-4 - (-4)} = 12^{-4 + 4} = 12^{0} \]

### Step 5: Evaluate \(12^{0}\)
Any non-zero number raised to the power of 0 is 1:
\[ 12^{0} = 1 \]

### Final Answer
Thus, the value of the expression \(\frac{(12)^{-2})^{2}}{(12)^{2})^{-2}}\) is:
\[ \boxed{1} \]