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)

Question

Responses

2−7⋅9−129⋅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

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

Answer 1

To simplify the expression \( (2^{-3} \cdot 9^3) - 4 \cdot 9^{-10} \) and express it with only positive exponents, we can follow these steps:

1. Rearrange the First Term: \[ 2^{-3} \cdot 9^3 = \frac{9^3}{2^3} = \frac{729}{8} \]

2. Convert the Second Term: \[ 4 \cdot 9^{-10} = 4 \cdot \frac{1}{9^{10}} = \frac{4}{9^{10}} = \frac{4}{9^{10}} = \frac{4}{3486784401} \quad \text{(not required to calculate at this stage)} \]

3. Combine Terms: The complete expression would now be: \[ \frac{9^3}{2^3} - \frac{4}{9^{10}} \]

This expression is still somewhat complex for our goal of having only positive exponents, so let’s look at the original expression in terms of properties of exponents.

Now let's assess the choices provided.

From the original expression: \[ (2^{-3} \cdot 9^3) - 4 \cdot 9^{-10} \]

Given options:

1. \( 2^{-7} \cdot 9^{-1} \)
2. \( \frac{2^3}{9^2} \)
3. \( 2^{12} \cdot 9^{-12} \)
4. \( 2^3 \cdot 9^2 \)

We can examine these options for equivalence in the context of the original expression and working towards positive exponents.

Notice the critical patterns:

1. \( 2^{-3} \) becomes \( \frac{1}{2^3} \) when using positive exponents.
2. \( 9^3 \) is fine as it is.
3. \( 9^{-10} \) can be converted to \( \frac{1}{9^{10}} \).

After computing the above insights and terms, none of the options provided match our derived forms from the original expression, suggesting a mistake in how the question or answers were prepared.

However, we can identify the one that follows the general trends or suggest an option closely.

Among the options:

• The second option \( \frac{2^3}{9^2} \) appears the closest to aligning due to the clarification in properties of exponent handling.

Thus, considering proper workings, the correct answer among the given options is: \[ \frac{2^3}{9^2} \] as the reduction suggests it pays to check each exponent transformation closely. But again, confirm with original context for correctness since mapping transformations can influence with deeper algebraic nuances not shown fully in detail here.