Short Answer 
The original expression (dfrac{1}{5} cdot dfrac{1}{5} cdot dfrac{1}{5} cdot dfrac{1}{5}) simplifies to (5^{-4}). Among the options provided, Options A ((5^{-2})^2) and D (5^{2} cdot 5^{-6}) are equivalent to (5^{-4}), while Options B and C are not.
Step 1: Understand the Given Expression
The expression we start with is (dfrac{1}{5} cdot dfrac{1}{5} cdot dfrac{1}{5} cdot dfrac{1}{5}). This can be simplified by recognizing that multiplying (dfrac{1}{5}) four times is the same as raising it to the fourth power of the denominator. Therefore, the expression can be rewritten as (dfrac{1}{5^4}) or 5^{-4}.
Step 2: Analyze Each Option for Equivalence
We will check each provided option to see if they are equivalent to (5^{-4}). The steps involved are:
- Option A: (5^{-2})^2 simplifies to 5^{-4}, which is equivalent!
- Option B: (5^{-4})^0 results in 5^{0} which is not equivalent!
- Option C: (dfrac{5^1}{5^4}) simplifies to 5^{-3}, so it is not equivalent!
- Option D: 5^{2} cdot 5^{-6} simplifies to 5^{-4}, making it equivalent!
Step 3: Identify the Correct Options
Based on our analysis, the options equivalent to the original expression (5^{-4}) are Option A and Option D. While options B and C do not provide equivalent values, confirming whether they match our expression is crucial in conclusion.