Which of the following is an equivalent expression to (2^−3 ⋅9^3)^−4/2^9 ⋅9^−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)

Question

Responses

2392
Start Fraction 2 cubed over 9 squared end fraction

212⋅9−1229⋅9−10
Start fraction 2 superscript 12 baseline times 9 superscript negative 12 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction

23 ⋅92
2 cubed times 9 squared

2−7⋅9−129⋅9−10

Answer 1

To simplify the expression \((2^{-3} \cdot 9^{3})^{-4} / (2^{9} \cdot 9^{-10})\) and find an equivalent expression with only positive exponents, we will use the properties of exponents step by step.

1. Apply the Power of a Product Rule: \[ (a \cdot b)^{-n} = a^{-n} \cdot b^{-n} \] Thus, we can distribute the \(-4\) across the terms inside the parentheses: \[ (2^{-3})^{-4} \cdot (9^{3})^{-4} = 2^{12} \cdot 9^{-12} \]

2. Rewrite the denominator: The expression now becomes: \[ \frac{2^{12} \cdot 9^{-12}}{2^{9} \cdot 9^{-10}} \]

3. Apply the Quotient Rule: Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify each base:

   • 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: Thus, the overall expression simplifies to: \[ 2^3 \cdot 9^{-2} \]

5. Convert to positive exponents: We can write \(9^{-2}\) as \(\frac{1}{9^2}\): \[ 2^3 \cdot 9^{-2} = \frac{2^3}{9^2} \]

Therefore, the simplified version of the expression with only positive exponents is: \[ \frac{2^3}{9^2} \]

Thus, the equivalent expression to \((2^{-3} \cdot 9^{3})^{-4} / (2^{9} \cdot 9^{-10})\), with only positive exponents, is: \[ \text{Start Fraction 2 cubed over 9 squared end fraction} \]

The correct answer is: \[ \frac{2^3}{9^2} \]