Question

which of the following is an equivalent expression to (2-3 .9^3)/2^9 .9-^10 with only positive exponents, generated by applying the properties of integer exponents?

2^12 .9-^12/2^9 .9-^10
2^3 . 9^2
2-^7 .9-^1/2^9 .9-^10
2^3/9^2

Answer 1

Let's simplify the expression \((2 - 3 \cdot 0.9^3) / (2^9 \cdot 0.9^{-10})\) and find an equivalent expression with only positive exponents.

1. Rewrite the expression with \( 0.9 \) as \( 9^{-1} \):

\[
(2 - 3 \cdot (9^{-1})^3) / (2^9 \cdot (9^{-1})^{-10})
\]

This simplifies to:

\[
(2 - 3 \cdot 9^{-3}) / (2^9 \cdot 9^{10})
\]

The term \( 3 \cdot 9^{-3} \) can be calculated or left as-is for now.

2. Now, combine the fractions:

\[
= \frac{2 - 3 \cdot 9^{-3}}{2^9 \cdot 9^{10}}
\]

To combine this, we need to express \( 2 \) with the common denominator of \( 9^{-3} \):

\[
= \frac{2 \cdot 9^3 - 3}{2^9 \cdot 9^{10}}
\]

3. Now simplifying the numerator, we have \( 2 \cdot 9^3 - 3 \), and when we express it back as a fraction, we will multiply it out by \( 9^{10} \).

4. As there is nothing in the expression that will cancel further on this numerator term with the existing expressions given above, we now need to check the simplified versions provided:

Now let's analyze the answer choices:

- **Choice A:** \( 2^{12} \cdot 9^{-12} / 2^9 \cdot 9^{-10} \)

This can be simplified as:
\[
= \frac{2^{12}}{2^9} \cdot \frac{9^{-12}}{9^{-10}} = 2^{12-9} \cdot 9^{-12 + 10} = 2^3 \cdot 9^{-2}
\]

- **Choice B:** \( 2^3 \cdot 9^2 \) does not align with our structure.
- **Choice C:** \( 2^{-7} \cdot 9^{-1} / 2^9 \cdot 9^{-10} \) gives an incorrect dynamic.

- **Choice D:** \( 2^3 / 9^2 \) also doesn't take the denominator structure into consideration.

**Comparing the final output from A:**

From option A: \( 2^3 \cdot 9^{-2} = 2^3 / 9^2\)

So the equivalent expression which can be expressed solely with positive exponents through required simplifications will conclude as choice:

\[
\boxed{2^3 / 9^2}
\]