Which expressions are equivalent to 
 
 
 
 
\[\dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}\] ?
Choose 2 answers:
Choose 2 answers:
(Choice A)   
\[(5^{-2})^{2}\]
A
\[(5^{-2})^{2}\]
(Choice B)   
\[(5^{-4})^{0}\]
B
\[(5^{-4})^{0}\]
(Choice C)   
 
\[\dfrac{5^1}{5^4}\]
C
 
\[\dfrac{5^1}{5^4}\]
(Choice D)   
\[5^2\cdot 5^{-6}\]
D
\[5^2\cdot 5^{-6}\]

Let's simplify the given expression first:

\[
\dfrac{1}{5} \cdot \dfrac{1}{5} \cdot \dfrac{1}{5} \cdot \dfrac{1}{5} = \dfrac{1}{5^4} = 5^{-4}
\]

Now, let's analyze the choices:

- **Choice A:** \((5^{-2})^2\)

Using the power of a power rule: \((a^m)^n = a^{m \cdot n}\), we have:

\[
(5^{-2})^2 = 5^{-2 \cdot 2} = 5^{-4}
\]

This matches the given expression, so **Choice A** is correct.

- **Choice B:** \((5^{-4})^0\)

Using the property that any number raised to the power of 0 is 1, we have:

\[
(5^{-4})^0 = 1
\]

This does not match \(5^{-4}\), so **Choice B** is incorrect.

- **Choice C:** \(\dfrac{5^1}{5^4}\)

Using the property of exponents \(\dfrac{a^m}{a^n} = a^{m-n}\), we get:

\[
\dfrac{5^1}{5^4} = 5^{1-4} = 5^{-3}
\]

This does not match \(5^{-4}\), so **Choice C** is incorrect.

- **Choice D:** \(5^2 \cdot 5^{-6}\)

Using the product of powers rule \(a^m \cdot a^n = a^{m+n}\), we get:

\[
5^2 \cdot 5^{-6} = 5^{2 + (-6)} = 5^{-4}
\]

This matches the given expression, so **Choice D** is correct.

### Conclusion:
The two correct choices are **A** and **D**.

Would you like further details or explanations?

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Here are 5 related questions to expand your understanding:

1. How does the power of a power rule work for negative exponents?
2. What are the properties of exponents when multiplying or dividing powers with the same base?
3. How do you simplify expressions involving negative exponents?
4. What happens when you raise any number to the power of 0?
5. How do you combine terms with exponents that involve both multiplication and division?

**Tip:** Always double-check the properties of exponents, especially when working with negative or fractional powers!

Which expressions are equivalent to 
\[
\dfrac{1}{5} \cdot \dfrac{1}{5} \cdot \dfrac{1}{5} \cdot \dfrac{1}{5} = \dfrac{1}{5^4} = 5^{-4}\]? Choose 2 answers: (Choice A) \[(5^{-2})^{2}\], (Choice B) \[(5^{-4})^{0}\], (Choice C) \[\dfrac{5^1}{5^4}\], (Choice D) \[5^2 \cdot 5^{-6}\].

Explore which expressions are equivalent to 5^{-4} by applying exponent rules, including the power of a power and product of powers rules. This problem provides practice in simplifying expressions and is suitable for students in Grades 8-10.

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