`(2^(-3))^(2)xx(3^(-2))^(3)=`_____________

Options

A. `5^(-6)`
B. `6^(6)`
C. `6^(-6)`
D. none of these

Answer

To solve the expression \((2^{-3})^{2} \times (3^{-2})^{3}\), we will follow these steps:

### Step 1: Apply the power of a power rule
The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). We will apply this rule to both parts of the expression.

\[ (2^{-3})^{2} = 2^{-3 \cdot 2} = 2^{-6} \]
\[ (3^{-2})^{3} = 3^{-2 \cdot 3} = 3^{-6} \]

### Step 2: Combine the results
Now we can combine the results from Step 1:

\[ 2^{-6} \times 3^{-6} \]

### Step 3: Use the property of exponents
We can use the property that states \(a^{-n} = \frac{1}{a^n}\). Thus, we can rewrite our expression:

\[ 2^{-6} \times 3^{-6} = \frac{1}{2^6} \times \frac{1}{3^6} = \frac{1}{2^6 \times 3^6} \]

### Step 4: Combine the bases
Using the property of exponents that states \((a \times b)^n = a^n \times b^n\), we can express the denominator:

\[ \frac{1}{(2 \times 3)^6} = \frac{1}{6^6} \]

### Final Answer
Thus, the final answer is:

\[ \frac{1}{6^6} \]

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