Question

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

2 to the power of 3 ⋅9 to the power of 2
2 cubed times 9 squared

2 to the power of 12 ⋅9 to the power of −12 over 2 to the power of 9 ⋅9 to the power of −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

2 to the power of 3 over 9 to the power of 2
Start Fraction 2 cubed over 9 squared end fraction

2 to the power of −7 ⋅9 to the power of −1 over 2 to the power of 9 ⋅9 to the power of −10

Answer 1

Let's simplify the expression \((2^{-3} \cdot 9^3)^{-4}\) over \(2^9 \cdot 9^{-10}\) step by step while ensuring we end up with only positive exponents.

1. Start with the numerator:
\[
(2^{-3} \cdot 9^3)^{-4} = 2^{-3 \cdot -4} \cdot 9^{3 \cdot -4} = 2^{12} \cdot 9^{-12}
\]

2. The denominator remains:
\[
2^9 \cdot 9^{-10}
\]

3. Now combine the numerator and denominator:
\[
\frac{2^{12} \cdot 9^{-12}}{2^9 \cdot 9^{-10}} = \frac{2^{12}}{2^9} \cdot \frac{9^{-12}}{9^{-10}}
\]

4. Apply the properties of exponents:
- For the base \(2\):
\[
\frac{2^{12}}{2^9} = 2^{12 - 9} = 2^3
\]
- For the base \(9\):
\[
\frac{9^{-12}}{9^{-10}} = 9^{-12 - (-10)} = 9^{-12 + 10} = 9^{-2}
\]

5. Therefore, we have:
\[
2^3 \cdot 9^{-2} = \frac{2^3}{9^2}
\]

Since the prompt requests an expression with only positive exponents, the equivalent expression is:
\[
\frac{2^3}{9^2}
\]

Thus, the correct response is:
**2 to the power of 3 over 9 to the power of 2 (Start Fraction 2 cubed over 9 squared end fraction)**.