The problem asks which expressions are equivalent to:

$\frac{2^5}{6^5}$

Let’s simplify and analyze the given options:

1. Original Expression: $\frac{2^5}{6^5} = \left(\frac{2}{6}\right)^5 = \left(\frac{1}{3}\right)^5$

So, the original expression is $\left(\frac{1}{3}\right)^5$.

Now let's examine each option:

• A: $\frac{1}{3}$
  This is not equivalent to $\left(\frac{1}{3}\right)^5$, so A is not correct.

• B: $3^{-5}$
  This is equivalent to $\left(\frac{1}{3}\right)^5$, so B is correct.

• C: $(-4)^{-5}$
  This is not related to $\frac{2^5}{6^5}$, so C is not correct.

• D: $2^5 \cdot 6^{-5}$
  This can be rewritten as: $\frac{2^5}{6^5} = \left(\frac{1}{3}\right)^5$ Therefore, D is correct.

Conclusion:

The correct answers are B and D.

Would you like more details on any of the steps?
Here are 5 related questions you can explore:

1. How do you simplify fractions with powers in the numerator and denominator?
2. Why is $3^{-5}$ equivalent to $\left(\frac{1}{3}\right)^5$?
3. How does changing the base of exponents affect the result?
4. What rules govern negative exponents and their transformations?
5. Can negative bases with exponents ever be equivalent to positive ones?

Tip: When dealing with powers and fractions, always try simplifying the fraction first before dealing with exponents!