Which expressions are equivalent to

To simplify the expression \(\dfrac{2^5}{6^5}\), we can use the property of exponents to rewrite the denominator:

\[
\dfrac{2^5}{6^5} = 2^5 \cdot (6^{-5}) = 2^5 \cdot 6^{-5}
\]

So, option D is equivalent to \(\dfrac{2^5}{6^5}\).

Next, we can express \(6\) in terms of its prime factors:

\[
6 = 2 \cdot 3
\]

Thus,

\[
6^5 = (2 \cdot 3)^5 = 2^5 \cdot 3^5
\]

Now, substituting this back into our expression:

\[
\dfrac{2^5}{6^5} = \dfrac{2^5}{2^5 \cdot 3^5}
\]

This simplifies to:

\[
\dfrac{1}{3^5} = 3^{-5}
\]

So, option B is also equivalent to \(\dfrac{2^5}{6^5}\).

Now we can summarize our findings:

- Option A: \(\dfrac{1}{3}\) is NOT correct because \(\dfrac{2^5}{6^5} = 3^{-5}\), not \(\dfrac{1}{3}\).
- Option B: \(3^{-5}\) is correct.
- Option C: \((-4)^{-5}\) is NOT a correct equivalence, because it is not related to the expression we started with.
- Option D: \(2^5 \cdot 6^{-5}\) is correct.

Thus, the two correct answers are:

- (Choice B) \(3^{-5}\)
- (Choice D) \(2^5 \cdot 6^{-5}\)