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)

Question

Responses

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

2^3 ⋅9^2
2 cubed times 9 squared

2^−7⋅9^−1/2^9⋅9^−10

Answer 1

To simplify the expression \( (2^{-3} \cdot 9^{3})^{-4} / (2^{9} \cdot 9^{-10}) \) and find an equivalent expression with only positive exponents, we'll first simplify the numerator using the properties of exponents.

Starting with the expression:

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

Using the power of a product property \((a^m \cdot b^m = a^{m} \cdot b^{n})\)

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

Using the power of a power property \((a^{m})^{n} = a^{mn}\)

\[ = 2^{12} \cdot 9^{-12} \]

Now, substituting this back into the entire expression gives us:

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

Now, using the quotient of powers property \(\frac{a^m}{a^n} = a^{m-n}\):

\[ = 2^{12-9} \cdot 9^{-12 - (-10)} = 2^{3} \cdot 9^{-12 + 10} = 2^{3} \cdot 9^{-2} \]

Next, to express this with only positive exponents, we use the fact that \(a^{-m} = \frac{1}{a^m}\):

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

Thus, the expression simplifies to:

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

So the equivalent expression to \((2^{-3} \cdot 9^{3})^{-4} / (2^{9} \cdot 9^{-10})\) with only positive exponents is:

\(\frac{2^{3}}{9^{2}}\) (which can be written as \(2^3 / 9^2\)).

The correct response is:

\(2^{3}/9^{2}\)