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) \(2^5 \cdot 6^{-5}\)