Which of the following is an equivalent expression to (2^−3 ⋅9^3)^−4/2^9 ⋅9^−10

Question

with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
Responses

Answer 1

To simplify the expression \((2^{-3} \cdot 9^3)^{-4} / (2^9 \cdot 9^{-10})\) and express it with only positive exponents, we can follow these steps:

1. Apply the exponent to the terms in parentheses: \[ (2^{-3} \cdot 9^3)^{-4} = (2^{-3})^{-4} \cdot (9^3)^{-4} = 2^{12} \cdot 9^{-12} \]

2. Substitute back into the expression: \[ \frac{2^{12} \cdot 9^{-12}}{2^9 \cdot 9^{-10}} \]

3. Use the properties of exponents to simplify the fraction:

   • For the base \(2\): \[ \frac{2^{12}}{2^9} = 2^{12 - 9} = 2^3 \]

   • For the base \(9\): \[ \frac{9^{-12}}{9^{-10}} = 9^{-12 - (-10)} = 9^{-12 + 10} = 9^{-2} \]

4. Combine the results: \[ 2^3 \cdot 9^{-2} \]

5. Convert \(9^{-2}\) to use positive exponents: \[ 9^{-2} = \frac{1}{9^2} \]

6. Write the expression with only positive exponents: \[ 2^3 \cdot \frac{1}{9^2} = \frac{2^3}{9^2} \]

7. Final Result: \[ \frac{8}{81} \quad (\text{since } 2^3 = 8 \text{ and } 9^2 = 81) \]

Thus, the equivalent expression with only positive exponents is:

Answer: \(\frac{8}{81}\)