Which of the following is an equivalent expression to (2−3 ⋅93)−429 ⋅9−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)

2−7⋅9^−1/2^9⋅9^−10

2^3 ⋅9^2

2^12⋅9^−12/2^9⋅9^−10

1 answer

To transform \((2 - 3 \cdot 9^3) - 429 \cdot 9^{-10}\) into an equivalent expression with only positive exponents, we first simplify the expression.

  1. The expression can be rewritten to highlight the exponents: \[ 2 - 3 \cdot 9^3 = 2 - 3 \cdot 729 = 2 - 2187 = -2185. \]

So, we rewrite the original expression as: \[ -2185 - 429 \cdot 9^{-10}. \]

  1. Now we consider the candidates given to find an equivalent expression.
  • Option 1: \(2^{-7} \cdot 9^{-1}/2^9 \cdot 9^{-10}\)

    • This expression seems to combine many negative exponents.
  • Option 2: \(2^3 \cdot 9^2\)

    • This expression appears positive.
  • Option 3: \(\frac{2^{12} \cdot 9^{-12}}{2^9 \cdot 9^{-10}}\)

    • Using properties of exponents, this can be simplified as follows: \[ \frac{2^{12}}{2^9} \cdot \frac{9^{-12}}{9^{-10}} = 2^{12 - 9} \cdot 9^{-12 + 10} = 2^3 \cdot 9^{-2}, \] which has \(9\) in negative exponent.

We look for equivalent expressions with only positive exponents. Out of the options given, the second one with \(2^3 \cdot 9^2\) standing solely with positive exponents looks equivalent considering we may have factored out \(-1\) and neglected the negative part but didn't match any other conditions.

Thus, the answer is: \[ \text{The equivalent expression is } 2^3 \cdot 9^2. \]