Apply All Exponent Rules Quick Check

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Question
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)
Responses

2^3 ⋅9^2
2 cubed times 9 squared

2^3/9^2
Start Fraction 2 cubed over 9 squared end fraction

2^12⋅9^−12/2^9⋅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

2^−7⋅9^−1/2^9⋅9^−10
Start fraction 2 superscript negative 7 baseline times 9 superscript negative 1 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
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Answer 1

To simplify the expression \((2^{-3} \cdot 9^3)^{-4} / (2^9 \cdot 9^{-10})\) and express it using only positive exponents, let's go through the steps by applying exponent rules:

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

2. Now substitute back into the expression: \[ 2^{12} \cdot 9^{-12} / (2^9 \cdot 9^{-10}) \]

   This can be written as: \[ \frac{2^{12} \cdot 9^{-12}}{2^9 \cdot 9^{-10}} \]

3. Apply the quotient rule of exponents (subtract the exponent in the denominator from the exponent in the numerator): \[ = 2^{12 - 9} \cdot 9^{-12 - (-10)} = 2^{3} \cdot 9^{-12 + 10} = 2^3 \cdot 9^{-2} \]

4. Convert to positive exponents: \[ 2^3 \cdot \frac{1}{9^2} = \frac{2^3}{9^2} \]

So, the expression can be rewritten as: \[ \frac{2^3}{9^2} \]

The correct answer choice is: \[ \text{2}^3/\text{9}^2 = \text{Start Fraction 2 cubed over 9 squared end Fraction} \]