Which of the following is an equivalent expression to \( \frac{\left(2^{-3} \cdot 9^{3}\right)^{-4}}{2^{9} \cdot 9^{-10}} \) with only positive exponents, generated by applying the Properties of Integer Exponents? (1 point) \( \frac{2^{3}}{9^{2}} \)

To simplify the expression \( \frac{\left(2^{-3} \cdot 9^{3}\right)^{-4}}{2^{9} \cdot 9^{-10}} \), we will apply the properties of integer exponents step by step. 1. **Simplify the numerator**: \[ \left(2^{-3} \cdot 9^{3}\right)^{-4} = (2^{-3})^{-4} \cdot (9^{3})^{-4} \] Using the property \( (a^m)^n = a^{m \cdot n} \): \[ (2^{-3})^{-4} = 2^{12} \quad \text{and} \quad (9^{3})^{-4} = 9^{-12} \] Therefore, the numerator becomes: \[ 2^{12} \cdot 9^{-12} \] 2. **Combine the numerator and denominator**: Now we have: \[ \frac{2^{12} \cdot 9^{-12}}{2^{9} \cdot 9^{-10}} \] 3. **Separate the fractions**: We can separate the fractions: \[ \frac{2^{12}}{2^{9}} \cdot \frac{9^{-12}}{9^{-10}} \] 4. **Apply the properties of exponents**: For the first fraction: \[ \frac{2^{12}}{2^{9}} = 2^{12 - 9} = 2^{3} \] For the second fraction: \[ \frac{9^{-12}}{9^{-10}} = 9^{-12 - (-10)} = 9^{-12 + 10} = 9^{-2} \] 5. **Combine the results**: Now we have: \[ 2^{3} \cdot 9^{-2} \] 6. **Convert to positive exponents**: To express \( 9^{-2} \) with a positive exponent, we can write: \[ 9^{-2} = \frac{1}{9^{2}} \] Thus, the expression becomes: \[ \frac{2^{3}}{9^{2}} \] Therefore, the equivalent expression to \( \frac{\left(2^{-3} \cdot 9^{3}\right)^{-4}}{2^{9} \cdot 9^{-10}} \) with only positive exponents is: \[ \frac{2^{3}}{9^{2}} \] So the answer is: \[ \frac{2^{3}}{9^{2}} \]