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 \((2^{-3} \cdot 9^{3})^{-4}/(2^{9} \cdot 9^{-10})\) using the properties of integer exponents, we will follow these steps:

1. Apply the Negative Exponent Rule: When raising a product to a power, we can distribute the exponent to each factor.

   \[ (2^{-3} \cdot 9^3)^{-4} = 2^{-3 \cdot -4} \cdot 9^{3 \cdot -4} = 2^{12} \cdot 9^{-12} \]

2. Write out the whole expression: Now we rewrite the entire expression including the denominator:

   \[ \frac{2^{12} \cdot 9^{-12}}{2^{9} \cdot 9^{-10}} \]

3. Apply the Quotient Rule: The quotient rule states that for any bases \(a\), \(m\), and \(n\):

   \[ \frac{a^m}{a^n} = a^{m-n} \]

   Apply this rule to simplify each base:

   • For \(2\):

   \[ \frac{2^{12}}{2^{9}} = 2^{12 - 9} = 2^{3} \]

   • For \(9\):

   \[ \frac{9^{-12}}{9^{-10}} = 9^{-12 - (-10)} = 9^{-12 + 10} = 9^{-2} \]

4. Combine the Results: Now we have:

   \[ 2^{3} \cdot 9^{-2} \]

5. Convert Expressions with Negative Exponents: To express everything with positive exponents:

   \[ 2^{3} \cdot \frac{1}{9^2} = \frac{2^{3}}{9^{2}} \]

Thus, the equivalent expression with only positive exponents is:

\[ \frac{2^3}{9^2} \]

This is the final simplified expression.